Critical Markov Branching-Immigration Systems: Complete Classification of the State Space and Exact Rates of Convergence

1. Introduction

Branching systems are among the oldest and most persistent models of population growth, and the continuous-time Markov branching system is the member of the family in which the interplay between reproduction and time is the most transparent: an individual alive at time t waits an exponential time and is then replaced by a random number of offspring, independently of everything else. Throughout the paper t is time and runs over the axis $\mathcal{T}:=[0,\infty )$, while s is the argument of a generating function and runs over $[0,1)$; both are fixed formally in Section 2, and s is allowed to be complex only in Section 9 and in the numerical check of its bounds in Section 10. Adding an immigration stream, so that new individuals enter the population at the epochs of an independent Poisson-type mechanism, turns the model into a genuinely different object. The population can no longer become extinct in the absolute sense, and the question of long-time behaviour becomes a question about the recurrence structure of an irreducible Markov chain on the nonnegative integers. Classical accounts are given in [3,14,36], and the immigration model itself goes back to Sevastyanov [35], Yang [40] and Pakes [30]; in discrete time see Foster and Williamson [10] and Zubkov [42], and, for the process stopped at the origin, Ivanoff and Seneta [21].

The critical case, in which the mean offspring number per individual is exactly one, is at the same time the most delicate and the most informative. Under a second moment condition on the offspring law and a first moment condition on the immigration law, Sevastyanov [35] and Pakes [30] showed that the transition functions decay polynomially and that a limiting generating function exists after multiplication by $t^{\lambda }$, where $\lambda$ is the ratio of the immigration mean to half the offspring variance. The power series expansion of that limit generates an invariant measure; for the system without immigration the corresponding structure is the theorem of Kesten, Ney and Spitzer [23]. Li, Chen and Pakes [25] refined the picture considerably, and Imomov [15] added rates of convergence under finiteness of the third derivative of the offspring generating function and of the second derivative of the immigration one.

Once the second moment of the offspring law is infinite the classical normalisation collapses, because the constant $\lambda$ that drives it is no longer defined. The natural replacement is a regular variation hypothesis: the offspring generating function $f\left(s\right)$ and the immigration generating function $h\left(s\right)$ are taken regularly varying at the point 1, of orders $1+\nu$ and $\delta$ with $\nu ,\delta \in (0,1]$ and with slowly varying factors $L(*)$ and $\ell (*)$. These are the hypotheses $\left[f_{\nu }\right]$, $\left[h_{\delta }\right]$ of Section 2, where both functions are written out in full; the classical case is $\nu =\delta =1$. Besides the two orders, one number built from the slowly varying factors will matter throughout, the limit ${\mathsf{\Lambda }}_{\infty }:={lim}_{x\to \infty }\ell \left(x\right)/L\left(x\right)$, introduced formally at (13). In discrete time, limit theorems for the critical process with immigration and infinite offspring variance go back to Pakes [31], and regularly varying immigration has been taken up more recently in [4,5]. This exact setting was worked out in [18,25], where it was shown that the sign of the single parameter $\gamma =\delta -\nu$ separates a recurrent from a transient regime, and four of the present authors [17] developed the transient side of that dichotomy in detail; rates of convergence in the two regimes $\gamma \ne 0$ were taken up in [16]. That line of work leaves open the boundary between the two regimes, the normalisation of the transient one and the control of the slowly varying factors, and the present paper answers all three.

Nothing is said there about the boundary $\gamma =0$, which is precisely the stratum containing the classical case $\nu =\delta =1$; and the classical case is known to split further, into a recurrent and a transient part, according to the size of $\lambda$. One expects a dichotomy on the whole of the boundary, and we prove that there is one: for $\gamma =0$ the state space is null recurrent when ${\mathsf{\Lambda }}_{\infty }\le \nu$ and transient when ${\mathsf{\Lambda }}_{\infty }>\nu$, with ${\mathsf{\Lambda }}_{\infty }$ as above. At $\nu =\delta =1$ this reduces exactly to the classical threshold. Together with the two open regimes this gives a complete classification of the state space, stated as Theorem 6 below.

In the transient regime it is the normalisation that has to be corrected. When $\gamma <0$ the transition functions decay at a stretched exponential rate, and the correct normalising factor is $exp\left\{\mathsf{\Theta }\right(t\left)\right\}$ with

$$\mathsf{\Theta }\left(t\right)=({\mathsf{\Lambda }}_{\infty }/\left|\gamma \right|){\left[\tau \left(t\right)\right]}^{\left|\gamma \right|},$$

where $\tau \left(t\right)=1/q\left(t\right)$ is the reciprocal of the survival probability $q\left(t\right)$ of the branching system without immigration. With the simpler factor ${\left[\tau \left(t\right)\right]}^{\left|\gamma \right|}$ used there, the limit exists only under the additional hypothesis ${\mathsf{\Lambda }}_{\infty }=\left|\gamma \right|$, which cuts a null set out of the parameter space. Carrying the constant ${\mathsf{\Lambda }}_{\infty }$ through the normalisation costs nothing and removes the hypothesis altogether; this is Theorem 7.

The slowly varying factors themselves call for a control of a different kind. The standard device is to postulate a remainder in the shift relation $L\left(\mu t\right)/L\left(t\right)=1+w_{\mu }\left(t\right)$ and then to suppress the dependence of $w_{\mu }$ on the shift $\mu$. The suppression has a cost. All of the estimates in this circle of problems are driven not by $w_{\mu }$ but by the index function ${\epsilon }_L\left(x\right)=x{L}^{\prime }\left(x\right)/L\left(x\right)$ of the representation theorem, and the two are proportional with a factor that depends on $\mu$ and is not identically equal to one. We therefore state the hypotheses directly in terms of the index function, and record the relation to the shift remainder as a lemma. The gain is not only cleanliness. The identity that the index function satisfies is exact, and the exact identity, given as Lemma 3 below, produces a three-term asymptotic expansion of the survival probability,

$${\displaystyle \frac{1}{q^{\nu }\left(t\right)}}=\nu L_{\infty }t-{\displaystyle \frac{c_L}{\nu }}ln\left(\nu L_{\infty }t\right)+\mathfrak{c}+o\left(1\right)\mathrm{as}t\to \infty ,$$

in which $L_{\infty }$ is the limit of the slowly varying factor $L\left(x\right)$ of the offspring generating function, $c_L$ is the limit of $x^{\nu }$ times its index function, and $\mathfrak{c}$ is a constant built from the same data; all three are written out in terms of $f\left(s\right)$ alone in Theorem 2. When $L(*)$ is constant the exact identity behind this expansion degenerates to $1/\mathcal{V}\left(q\left(t\right)\right)=\nu t+1/a_0$, where $\mathcal{V}\left(y\right):=y^{\nu }L(1/y)$ is the auxiliary function introduced at (12) below, which is both elementary and classical; it is for a genuinely varying $L(*)$, including a non-constant one with $\nu =1$, that the expansion has content. A two-term form of it, with a logarithmic correction of the same shape but retaining an implicitly defined norming function, is [17, Thm. 1]; the coefficient carried there is the present one at $c_L=L_{\infty }$, which is a hypersurface of the parameter space and not an identity, and Remark 2 says where the difference comes from. What is added here is the third term, the constant $\mathfrak{c}$, and the removal of the norming function, which is what makes the expansion usable as it stands.

Beside this line of work should be placed the recent paper of Mitov and Yanev [28], which carries the same two regular variation hypotheses into a wider model: immigration there occurs at the jump epochs of a Poisson process of intensity $r\left(t\right)$ with $r\left(t\right)$ tending to a positive constant, so that the present system is the case of an intensity constant throughout, their offspring order being our $\nu$ and their immigration order our $\delta$. Their companion papers [27,29] treat the same model with an increasing intensity and with one tending to zero, and the present system is a case of neither. The results of [28] touch ours at four places, and only two of them concern a statement proved below, in each case on a stratum that was already known. Their criterion separates $\delta >\nu$ from $\delta \le \nu$ by the convergence of an integral, which is the positive recurrent stratum of Theorem 6(i); the splitting of the boundary $\delta =\nu$ into a null recurrent and a transient part is not addressed there, and it is that splitting which Theorem 6(ii) supplies. On that same boundary they obtain a proper limit law for the population size divided by $\tau \left(t\right)$, of Laplace transform ${(1+u^{\nu })}^{-{\mathsf{\Lambda }}_{\infty }/\nu }$ for $u>0$, and their divergent case covers the transient stratum $\delta <\nu$ as well, yielding there a stable limit law for the population size under its own normalisation. These two limit laws are statements in which the argument of the generating function approaches 1 as t grows, and they are complementary to Theorems 10 and 7 below, which keep the argument fixed and describe the invariant measure. Their convergent case yields for $\delta >\nu$ the limiting generating function of Theorem 11 below, without a rate; that limit is not new here either, and the head of Section 8 says where it does come from. The one regime of [28] that does not arise here at all is that in which the ratio $\ell \left(x\right)/L\left(x\right)$ of the two slowly varying factors degenerates: the standing hypotheses of Section 2 force its limit ${\mathsf{\Lambda }}_{\infty }$ to be finite and positive.

The rest of the paper is arranged as follows. Section 2 describes the model, fixes the Karamata classes in Section 2.2 and states the standing hypotheses. Section 3 is devoted to the exact identity and to the Basic Lemma of the critical branching system that follows from it, together with the survival probability, the local probability ${\mathsf{p}}_1\left(t\right)$ of finding that system in state 1, and the generating function of the invariant measure of the branching system without immigration. Section 4 stays with that system and gives rates for the conditioned transition functions and for the local ratios. Section 5 returns to the system with immigration and contains the classification theorem, and Section 6, Section 7 and Section 8 treat, in turn, the transient regime $\gamma <0$, the boundary $\gamma =0$ and the positive recurrent regime $\gamma >0$; each of them identifies the limiting generating function and the exact rate of approach to it. Section 9 carries every rate of the paper from the generating functions to the individual transition probabilities, by a complex-analytic argument resting on a factorisation that criticality forces and needing no monotonicity hypothesis. Section 10 carries out a numerical verification on explicit models in which every constant of the paper can be computed independently. Section 11 collects concluding remarks and the questions that remain open.

It remains to say how the present paper stands to the treatment of [17], of which it is the direct continuation. Section 2, Section 3 and Section 4 revisit material treated there; we have retained only what the new results need, and in each case the statement given here supersedes its earlier counterpart. The hypotheses are now placed on the index functions of the two slowly varying factors rather than on their shift remainders; the two formulations are related by Lemma 2, and the constants of the earlier one are to be read through the factor $\nu /(1-{\mu }^{-\nu })$ appearing there. The implicitly defined normalising function $\mathcal{N}(*)$ of that treatment is superfluous under the present hypotheses and has been replaced throughout by the explicit constant $1/L_{\infty }^{1/\nu }$; see Remark 2. The normalisation of the transient regime carries the constant ${\mathsf{\Lambda }}_{\infty }$, so that the hypothesis ${\mathsf{\Lambda }}_{\infty }=\left|\gamma \right|$ of the earlier version is not needed.

The generating function $M\left(s\right)$ of the invariant measure of the system without immigration is given here by Corollary 1. Its last term is absent from the expression stated as [17, Thm. 7], which therefore holds as an identity precisely when $L(*)$ is constant, and in particular in the classical finite variance case, where the distinction does not arise; for a genuinely varying $L(*)$ the two differ by a factor of the order of unity which tends to 1 only as $s\uparrow 1$, as Section 10 shows. The statements of Section 4 that rest on $M\left(s\right)$ are restated accordingly. The range of the indices is also wider, both being allowed to take the value 1: the earlier treatment assumes $\nu ,\delta \in (0,1)$, so that the classical finite variance case $\nu =\delta =1$ lies outside its range, whereas here it is a point of the boundary stratum and is covered by Theorem 10.

Nor is any normalisation imposed on the constants. In that treatment the shift remainders of the two slowly varying factors are written $\omega (*)$ and $\psi (*)$, the two constants ${\lambda }_{\omega }:={lim}_{x\to \infty }{x}^{\nu }\omega \left(x\right)$ and ${\lambda }_{\psi }:={lim}_{x\to \infty }{x}^{\delta }\psi \left(x\right)$ are formed from them, and the constant that governs the transient regime is their ratio ${\lambda }_{\psi }/{\lambda }_{\omega }$, on which the hypothesis of that regime is imposed. The constant that governs the same regime here is ${\mathsf{\Lambda }}_{\infty }$ of (13), the ratio of the two limits ${\ell }_{\infty }$ and $L_{\infty }$. The two agree, but only because the remainders are narrowed there by $x^{\nu }\omega \left(x\right)/L\left(x\right)\to 1$ as $x\to \infty$ and its analogue for $\psi (*)$, which force ${\lambda }_{\omega }=L_{\infty }$ and ${\lambda }_{\psi }={\ell }_{\infty }$. That these are hypotheses and not identities is what (10) below shows: it gives

$${\lambda }_{\omega }={\displaystyle \frac{c_L}{\nu }}(1-{\mu }^{-\nu })\mathrm{and}{\lambda }_{\psi }={\displaystyle \frac{c_{\ell }}{\delta }}(1-{\mu }^{-\delta }),$$

the second by the analogue of (10) for $\ell (*)$ under $\left(\mathcal{B}\right)$. Each narrowing is therefore a relation between two of our four constants, depending on the shift $\mu$ besides, and together they cut a set of codimension two out of the parameter space. Under the Basic Axioms nothing of the kind is assumed: $c_L$, $c_{\ell }$, $L_{\infty }$ and ${\ell }_{\infty }$ are independent and enter the results separately, and the ratio ${\lambda }_{\psi }/{\lambda }_{\omega }$ is then not ${\mathsf{\Lambda }}_{\infty }$.

Section 5, Section 7, Section 8, Section 9 and Section 10 have no counterpart in [17]. Some of them, and also Section 4, have one elsewhere, and the boundary between what is taken over and what is added should be said plainly. The classification of Section 5 is the evaluation, under the standing hypotheses, of a criterion that is classical and is recalled at (52) below; its content is the boundary stratum, the two other strata being the dichotomy already read off from that criterion in [18,25]. The limiting generating function of the positive recurrent regime of Section 8 is that of [18, Thm. 1.2], and the rate of approach to it, with its constant, is [16, Thm. 2.1]; what is added there is a rate expressed in constants rather than in slowly varying factors, the monotonicity and the uniformity of the approach, and the passage to the transition probabilities. The two limits of Section 4, finally, are, up to notation, [17, Thms. 8, 9], with $M\left(s\right)$ there in the reduced form that Corollary 1 corrects; what is added there is the second-order correction of Theorem 4 below, its uniformity in s, and the transfer to the individual probabilities. The head of each of the three sections points back to here.

2. The Model and the Standing Hypotheses

2.1. The Branching-Immigration System

Let $\mathbb{N}$ be the set of natural numbers, let ${\mathbb{N}}_0:=\left\{0\right\}\cup \mathbb{N}$ and let $\mathcal{T}:=[0,\infty )$ be the time axis. We consider a population of monotype individuals. Each individual alive at time $t\in \mathcal{T}$, independently of its history and of all other individuals, is transformed during $(t,t+\varsigma )$ according to the rule that it produces $j\in {\mathbb{N}}_0\setminus \left\{1\right\}$ offspring with probability $a_j\varsigma +o\left(\varsigma \right)$, and with probability $1+a_1\varsigma +o\left(\varsigma \right)$ either survives without reproducing or produces exactly one offspring, all as $\varsigma \downarrow 0$. The branching rates $\{a_j,j\in {\mathbb{N}}_0\}$ satisfy $a_j\ge 0$ for $j\in {\mathbb{N}}_0\setminus \left\{1\right\}$ together with the consistency condition

$$0<a_0<-a_1=\sum _{j\in {\mathbb{N}}_0\setminus \left\{1\right\}}{a}_j.$$

Independently of the branching mechanism, $j\in \mathbb{N}$ new individuals enter the population during $(t,t+\varsigma )$ with probability $b_j\varsigma +o\left(\varsigma \right)$, and no immigration occurs with probability $1+b_0\varsigma +o\left(\varsigma \right)$, again as $\varsigma \downarrow 0$. The immigration intensities satisfy $b_j\ge 0$ for $j\in \mathbb{N}$ and $0<-b_0={\sum }_{k\in \mathbb{N}}{b}_k<\infty$. Immigrants reproduce according to the same branching rates. The two mechanisms are encoded by the infinitesimal generating functions

$$f\left(s\right):=\sum _{k\in {\mathbb{N}}_0}{a}_k{s}^k\mathrm{and}h\left(s\right):=\sum _{k\in {\mathbb{N}}_0}{b}_k{s}^k$$

for $s\in [0,1)$. Both vanish at $s=1$, and $f\left(0\right)=a_0>0$, $h\left(0\right)=b_0<0$. The resulting stochastic model is called the continuous-time Markov branching-immigration system, abbreviated MBIS. Its relation to the discrete-time model is the embedding problem of Karlin and McGregor [22].

Write $X\left(t\right)$ for the population size at time $t\in \mathcal{T}$. The process $\left\{X\right(t),t\in \mathcal{T}\}$ is a homogeneous continuous-time Markov chain on a state space $\mathcal{E}\subseteq {\mathbb{N}}_0$ with transition functions

$$p_{ij}\left(t\right):={\mathbb{P}}_i\{X\left(t\right)=j\}=\mathbb{P}\{X(u+t)=j|X\left(u\right)=i\}$$

for $i,j\in \mathcal{E}$ and $u,t\in \mathcal{T}$; here t is the length of the transition interval and not an infinitesimal quantity. We take $\mathcal{E}$ to be the set of states reachable from 0. Because $a_0>0$, every line of descent can die out, so that 0 is reachable from each state of $\mathcal{E}$; hence $\mathcal{E}$ is a single communicating class and the chain is irreducible on it. One has $\mathcal{E}={\mathbb{N}}_0$ as soon as $b_1>0$, and nothing below depends on which subset $\mathcal{E}$ is. The chain is also honest, that is non-explosive: the total jump rate of an individual is $\left|a_1\right|<\infty$, the total immigration rate is $\left|b_0\right|<\infty$, and in the critical case the branching part cannot explode; see [3, Ch. III] and [2, Ch. 2]. The generating function ${\mathcal{P}}_i(t;s):={\sum }_{j\in \mathcal{E}}{p}_{ij}\left(t\right)s^j$ is a power series with nonnegative coefficients of sum at most 1, so that it converges absolutely on the closed unit disc and is analytic on the open one, and the same holds for $F(t;s)$ below. The remaining generating functions of the paper have nonnegative coefficients as well, and each is shown analytic on $\left|s\right|<1$ where it is introduced. We nevertheless write the argument as a real $s\in [0,1)$ throughout, the sole exception being Section 9, where the extension to complex s is precisely what carries the results to the individual transition probabilities. The generating function admits the representation

$${\mathcal{P}}_i(t;s)={\left[F(t;s)\right]}^{i}exp\left\{{\int }_0^{t}h\left(F(u;s)\right)\mathbf{d}u\right\},$$

(1)

where $F(t;s)$ solves the backward

$${\displaystyle \frac{\partial F(t;s)}{\partial t}}=f\left(F(t;s)\right)$$

(2)

and the forward

$${\displaystyle \frac{\partial F(t;s)}{\partial t}}=f\left(s\right){\displaystyle \frac{\partial F(t;s)}{\partial s}}$$

(3)

Kolmogorov equations subject to $F(0;s)=s$; see [18,19,25,26,30,36].

Switching the immigration off amounts to $h\left(s\right)\equiv 0$, that is, to $b_k=0$ for every $k\in {\mathbb{N}}_0$. What remains is the Markov branching system without immigration, abbreviated MBS, whose population size we denote by $Z\left(t\right)$. It is a reducible chain on ${\mathcal{S}}_0=\left\{0\right\}\cup \mathcal{S}$ with $\mathcal{S}\subseteq \mathbb{N}$ a communicating class of transient states and $\left\{0\right\}$ absorbing, and

$$\sum _{j\in {\mathcal{S}}_0}{\mathsf{p}}_{ij}\left(t\right)s^j={\left[F(t;s)\right]}^i,\mathrm{where}F(t;s)=\sum _{k\in {\mathcal{S}}_0}{\mathsf{p}}_k\left(t\right)s^k$$

and ${\mathsf{p}}_k\left(t\right):=\mathbb{P}\{Z\left(t\right)=k|Z\left(0\right)=1\}$. Thus $F(t;s)$ is common to both systems, and (1) exhibits the MBIS as the immigration stream superposed on it. The extinction probability of the MBS is the smallest root of $f\left(s\right)=0$ in $[0,1]$, and $F(t;s)$ converges to it as $t\to \infty$, uniformly in $s\in [0,d]$ for each $d<1$; see [30].

Differentiating (1) at $s=1$ gives

$$\mathbb{E}\left[X\left(t\right)\right|X\left(0\right)=0]={\displaystyle \frac{\partial {\mathcal{P}}_0(t;s)}{\partial s}}{|}_{s=1}=\left\{\begin{array}{cc}{\displaystyle \frac{h^{\prime }(1-)}{a}}(e^{at}-1),\hfill & \mathrm{if}a\ne 0,\hfill \\ beginequation3mm]h^{\prime }(1-)t,\hfill & \mathrm{if}a=0,\hfill \end{array}\right.$$

(4)

where $a=f^{\prime }(1-)$ and, here and throughout, the argument $1-$ marks the limit as the variable increases to 1. Formula (4) shows that the growth of the mean is governed by the sign of a alone: the system is subcritical, critical or supercritical according as $a<0$, $a=0$ or $a>0$, and in the first two cases the extinction probability equals 1. Throughout the paper the system is critical, so that

$$f\left(1\right)=0\mathrm{and}{f}^{\prime }(1-)=0.$$

In particular $f\left(s\right)>0$ on $[0,1)$, because $f\left(s\right)$ is convex with $f\left(1\right)=f^{\prime }(1-)=0$; and $h\left(s\right)<0$ on $[0,1)$, because $h\left(s\right)$ is nondecreasing with $h\left(1\right)=0$ and $h\left(s\right)\neg \equiv 0$. Finally, (1) shows that ${\mathcal{P}}_i(t;s)/{\mathcal{P}}_0(t;s)\to 1$ as $t\to \infty$ for every $i\in \mathcal{E}$, the extinction probability being 1, so that it suffices throughout to study

$$\mathcal{P}(t;s):={\mathcal{P}}_0(t;s)=exp\left\{{\int }_0^{t}h\left(F(u;s)\right)\mathbf{d}u\right\},$$

whose value at $s=0$ is the return probability $p_{00}\left(t\right)$.

2.2. Karamata Classes and the Index Function

A positive measurable function $L(*)$ defined on a neighbourhood of infinity is slowly varying at infinity if $L\left(\mu x\right)/L\left(x\right)\to 1$ as $x\to \infty$ for every $\mu >0$; the class of such functions is denoted ${\mathcal{SV}}_{\infty }$. A function $V(*)$ is regularly varying at infinity with index $\rho$, written $V(*)\in {\mathcal{RV}}_{\infty }\left(\rho \right)$, if $V\left(x\right)=x^{\rho }L\left(x\right)$ for some $L(*)\in {\mathcal{SV}}_{\infty }$. The representation theorem states that every $L(*)\in {\mathcal{SV}}_{\infty }$ can be written as

$$L\left(x\right)=c\left(x\right)exp{\int }_{x_0}^x{\displaystyle \frac{\epsilon \left(u\right)}{u}}\mathbf{d}u$$

with $c\left(x\right)\to c_0\in (0,\infty )$ and $\epsilon \left(x\right)\to 0$ as $x\to \infty$; if $c(*)$ is constant then $L(*)$ is called normalised, and in that case $L(*)$ is differentiable with

$${\epsilon }_L\left(x\right):={\displaystyle \frac{x{L}^{\prime }\left(x\right)}{L\left(x\right)}}\mathrm{and}L\left(x\right)=L\left(x_0\right)exp{\int }_{x_0}^x{\displaystyle \frac{{\epsilon }_L\left(u\right)}{u}}\mathbf{d}u.$$

We call ${\epsilon }_L(*)$ the index function of $L(*)$. All slowly varying functions occurring below are normalised, so that the index function is defined; see [6, §1.3].

Most of the literature on branching systems controls the fine structure of $L(*)$ not through the index function but through a remainder in the shift relation, one writing $L\left(\mu x\right)/L\left(x\right)=1+w_{\mu }\left(x\right)$ and postulating a decay rate for $w_{\mu }(*)$, after which the subscript $\mu$ is suppressed and $w(*):=w_{\mu }(*)$ is treated as a single function; the subclass of ${\mathcal{SV}}_{\infty }$ so obtained is recalled at (7) below, where our own hypotheses are placed inside it. Both devices have been used, and the passage between them, with the classification of remainders it induces, is the subject of [20]. What is not harmless is the suppression of $\mu$: it is a notation and not an identification, since what the proofs actually use is the index function, and the two are proportional with a factor that depends on the shift and is not identically unity, as Lemma 2 below makes precise. We therefore place the hypotheses on the index function itself throughout.

2.3. Standing Hypotheses

The generating functions are assumed regularly varying at the point 1:

$$f\left(s\right)={(1-s)}^{1+\nu }L\left({\displaystyle \frac{1}{1-s}}\right)$$

[f_v]

and

$$h\left(s\right)=-{(1-s)}^{\delta }\ell \left({\displaystyle \frac{1}{1-s}}\right)$$

[h_δ]

for $s\in [0,1)$, where $\nu ,\delta \in (0,1]$ and $L(*),\ell (*)\in {\mathcal{SV}}_{\infty }$ are normalised. Hypothesis $\left[f_{\nu }\right]$ says that the offspring law lies in the domain of attraction of a $(1+\nu )$-stable law, the hypothesis under which the theory of the critical system without immigration has been developed since Slack [37] and Seneta [34]; see also Vatutin [39] and Pakes [33]. It forces $f^{\prime \prime }(1-)=\infty$ when $\nu <1$, while $\nu =1$ with $L\left(x\right)$ tending to a positive limit as $x\to \infty$ is the finite variance case, $f^{\prime \prime }(1-)$ being twice that limit. Hypothesis $\left[h_{\delta }\right]$ says that the immigration law has finite moments of every order below $\delta$ and, once $\ell (*)$ is bounded away from zero at infinity, as it is under $\left(\mathcal{B}\right)$ below, infinite moment of order $\delta$ when $\delta <1$, while $\delta =1$ with $\ell \left(x\right)\to h^{\prime }(1-)$ as $x\to \infty$ is the finite immigration mean case. Two parameters control everything that follows:

$$\gamma :=\delta -\nu \mathrm{and}\chi :=2\delta -\nu .$$

(5)

The fine structure of the two slowly varying factors is controlled by their index functions. We assume

$$x^{\nu }{\epsilon }_L\left(x\right)⟶c_L\in [0,\infty )\mathrm{as}x\to \infty$$

(A)

and

$$x^{\delta }{\epsilon }_{\ell }\left(x\right)⟶c_{\ell }\in [0,\infty )\mathrm{as}x\to \infty .$$

(B)

Where an exact remainder is needed we strengthen these to their quantitative forms

$$x^{\nu }{\epsilon }_L\left(x\right)=c_L+\mathcal{O}\left(x^{-{\rho }_L}\right)\mathrm{as}x\to \infty$$

(A*)

and

$$x^{\delta }{\epsilon }_{\ell }\left(x\right)=c_{\ell }+\mathcal{O}\left(x^{-{\rho }_{\ell }}\right)\mathrm{as}x\to \infty$$

(B*)

for some ${\rho }_L>0$ and ${\rho }_{\ell }>0$, and we write

$${\rho }_{*}:=min\{\nu ,\delta ,{\rho }_L,{\rho }_{\ell }\}$$

(6)

for the joint rate they produce. Hypotheses $\left[f_{\nu }\right]$, $\left[h_{\delta }\right]$, $\left(\mathcal{A}\right)$ and $\left(\mathcal{B}\right)$ together are referred to as the Basic Axioms.

The two constants $c_L$ and $c_{\ell }$ are allowed to vanish, and the case in which both do is not a degenerate one but the classical one: for $L(*)$ and $\ell (*)$ constant the index functions vanish identically, so that $c_L=c_{\ell }=0$, and $\left[f_{\nu }\right]$ at $\nu =\delta =1$ is the finite variance case. Nothing below is empty there: the expansion (29) below simply loses its logarithmic term, and the rate (75) below keeps the constant it has. Positivity is required at exactly two points of the paper, and is hypothesised there rather than here: $c_L>0$ in Lemma 2, where the index function is divided by the shift remainder, and $c_{\ell }>0$ in Theorem 7, where it is what makes the leading term of the transient rate the one displayed; what happens at $c_{\ell }=0$ is written out in Remark 7.

Hypotheses $\left({\mathcal{A}}^{*}\right)$ and $\left({\mathcal{B}}^{*}\right)$ are quantitative statements of membership in the Karamata subclass of slowly varying functions with remainder. Given a positive nonincreasing function $\alpha \left(x\right)$ with $\alpha \left(x\right)\to 0$ as $x\to \infty$, a function $L(*)\in {\mathcal{SV}}_{\infty }$ is said to be slowly varying with remainder $\alpha (*)$, written $L(*)\in {\mathcal{SR}}_{\infty }\left(\alpha \right)$, if

$${\displaystyle \frac{L\left(\mu x\right)}{L\left(x\right)}}=1+\mathcal{O}\left(\alpha \left(x\right)\right)\mathrm{as}x\to \infty$$

(7)

for every $\mu >0$; this is condition SR1 of [6, Ch. III,§3.12.1], the class and its calculus going back to Bojanic and Seneta [7], to Aljančić and Arandjelović [1] and to Goldie and Smith [12]; see also Geluk and de Haan [11]. The asymptotic theory of the critical branching system with infinite offspring variance has been developed precisely in this subclass, with the calibration $\alpha \left(x\right)=\mathcal{O}(L\left(x\right)/x^{\nu })$ for the offspring factor and $\alpha \left(x\right)=\mathcal{O}(\ell \left(x\right)/x^{\delta })$ for the immigration one; see [18, Thms. 1.2, 1.3] and the hypotheses preceding them. In that calibration, membership in ${\mathcal{SR}}_{\infty }\left(\alpha \right)$ is, by [6, Cor. 3.12.3], equivalent to $L\left(x\right)=L_{\infty }+\mathcal{O}\left(\alpha \left(x\right)\right)$ as $x\to \infty$, so that the finite positive limit is forced by the subclass itself and is not an extra assumption; Lemma 1 below obtains the same conclusion from the index function.

Our axioms lie inside that subclass, and Lemma 2 makes the inclusion explicit: $\left(\mathcal{A}\right)$ gives $L(*)\in {\mathcal{SR}}_{\infty }\left(x^{-\nu }\right)$ with the exact constant $c_L(1-{\mu }^{-\nu })/\nu$ in place of the $\mathcal{O}$ of (7), and the same holds for $\ell (*)$ with $\delta$ and $c_{\ell }$ in place of $\nu$ and $c_L$ under $\left(\mathcal{B}\right)$. Two things are gained by placing the hypotheses on the index function rather than on the remainder of (7). The remainder of (7) depends on the shift $\mu$ while the index function does not, so that every constant obtained below is shift-free; and the second-order rates ${\rho }_L$, ${\rho }_{\ell }$ of $\left({\mathcal{A}}^{*}\right)$ and $\left({\mathcal{B}}^{*}\right)$ have no counterpart in (7), whose $\mathcal{O}$ carries no constant at all; it is these two rates, through ${\rho }_{*}$ of (6), that turn every $o\left(1\right)$ of the paper into an explicit power of t.

It should be said at once what the Basic Axioms do and do not allow. By Lemma 1 below, hypothesis $\left(\mathcal{A}\right)$ forces $L\left(x\right)$ to have a finite positive limit, so that $\left[f_{\nu }\right]$ together with $\left(\mathcal{A}\right)$ is a condition of regular variation of the second order in the sense of de Haan and Stadtmüller [13],

$$f\left(s\right)={(1-s)}^{1+\nu }[L_{\infty }+\mathcal{O}\left({(1-s)}^{\nu }\right)]\mathrm{as}s\uparrow 1,$$

rather than a condition of plain regular variation: a factor such as $L\left(x\right)=lnx$ is excluded. What is retained is the essential feature of the problem, namely $f^{\prime \prime }(1-)=\infty$ for $\nu <1$, and what is gained is that every constant below is finite and explicit. The same remark applies to $\left[h_{\delta }\right]$ and $\left(\mathcal{B}\right)$. Hypotheses of this kind are exactly those under which the shift remainder of the earlier formulations decays at the rate $\mathcal{O}\left(x^{-\nu }\right)$ postulated there; see Lemma 2 and Remark 1.

Two consequences of the Basic Axioms are used constantly and are recorded at once.

Lemma 1. 

Let $\left(\mathcal{A}\right)$ hold. Then the limit $L_{\infty }:={lim}_{x\to \infty }L\left(x\right)$ exists in $(0,\infty )$ and

$$L_{\infty }-L\left(x\right)={\displaystyle \frac{L_{\infty }{c}_L}{\nu x^{\nu }}}+o\left(x^{-\nu }\right)\mathrm{as}x\to \infty .$$

(8)

If moreover $\left({\mathcal{A}}^{*}\right)$ holds, then the $o\left(x^{-\nu }\right)$ in (8) is $\mathcal{O}\left(x^{-\nu -min\{\nu ,{\rho }_L\}}\right)$. Finally

$$L^{\prime }\left(x\right)=\mathcal{O}\left(x^{-1-\nu }\right),\mathrm{so}\mathrm{that}L\left(y\right)-L\left(x\right)=\mathcal{O}\left((y-x)x^{-1-\nu }\right)$$

(9)

as $x\to \infty$, uniformly in $y\in [x,2x]$. The same statements hold for $\ell (*)$ with ν, $c_L$, ${\rho }_L$ replaced by δ, $c_{\ell }$, ${\rho }_{\ell }$, the limit being denoted ${\ell }_{\infty }$.

Proof. 

By $\left(\mathcal{A}\right)$ we have ${\epsilon }_L\left(u\right)=\mathcal{O}\left(u^{-\nu }\right)$ as $u\to \infty$, so that the integral

$${\int }_{x_0}^{\infty }[{\epsilon }_L\left(u\right)/u]\mathbf{d}u$$

converges absolutely. Since

$$L\left(x\right)=L\left(x_0\right)exp{\int }_{x_0}^x[{\epsilon }_L\left(u\right)/u]\mathbf{d}u,$$

the limit $L_{\infty }$ exists and is finite, and it is positive because the exponent is bounded. In the shift-remainder formulation the same conclusion is known; see [19]. Writing

$$L_{\infty }-L\left(x\right)=L_{\infty }(1-exp\{-{\int }_x^{\infty }{\displaystyle \frac{{\epsilon }_L\left(u\right)}{u}}\mathbf{d}u\})$$

and using $1-e^{-y}=y+\mathcal{O}\left(y^2\right)$ as $y\to 0$ together with

$${\int }_x^{\infty }{\displaystyle \frac{{\epsilon }_L\left(u\right)}{u}}\mathbf{d}u={\int }_x^{\infty }{\displaystyle \frac{u^{\nu }{\epsilon }_L\left(u\right)}{u^{1+\nu }}}\mathbf{d}u={\displaystyle \frac{c_L}{\nu x^{\nu }}}+o\left(x^{-\nu }\right)$$

gives (8); the quadratic term of $1-e^{-y}$ contributes $\mathcal{O}\left(x^{-2\nu }\right)$, which is $o\left(x^{-\nu }\right)$. The additive form is the one needed: the constant $c_L$ is allowed to vanish by $\left(\mathcal{A}\right)$, and a relative remainder would then assert that $L(*)$ is constant. Under $\left({\mathcal{A}}^{*}\right)$ the last display carries the remainder $\mathcal{O}\left(x^{-\nu -{\rho }_L}\right)$, and together with the quadratic term this is the asserted bound. Finally $L^{\prime }\left(x\right)=L\left(x\right){\epsilon }_L\left(x\right)/x$, and the first half of (9) follows because $L(*)$ is bounded and ${\epsilon }_L\left(x\right)=\mathcal{O}\left(x^{-\nu }\right)$ as $x\to \infty$; the second half is the first integrated over $[x,y]$, the integrand being of the constant order $x^{-1-\nu }$ throughout $[x,2x]$. □

Lemma 2. 

Let $\left(\mathcal{A}\right)$ hold with $c_L>0$ and put $w_{\mu }\left(x\right):=L\left(\mu x\right)/L\left(x\right)-1$. Then for every $\mu >0$

$$x^{\nu }{w}_{\mu }\left(x\right)⟶{\displaystyle \frac{c_L}{\nu }}(1-{\mu }^{-\nu })\mathrm{as}x\to \infty .$$

(10)

In particular $w_{\mu }\left(x\right)=\mathcal{O}\left(x^{-\nu }\right)$ as $x\to \infty$ for every $\mu >0$; relation (10), in which the limit is a function of μ, places $L(*)$ in the class $SR2$ of [6], treated for its own sake in [20]. Moreover

$${\displaystyle \frac{{\epsilon }_L\left(x\right)}{w_{\mu }\left(x\right)}}⟶{\displaystyle \frac{\nu }{1-{\mu }^{-\nu }}}\mathrm{as}x\to \infty .$$

(11)

Proof. 

Since

$$ln[1+w_{\mu }\left(x\right)]={\int }_x^{\mu x}[{\epsilon }_L\left(u\right)/u]\mathbf{d}u$$

and ${\epsilon }_L\left(u\right)=c_L{u}^{-\nu }(1+o\left(1\right))$ as $u\to \infty$, the integral equals $(c_L/\nu )x^{-\nu }(1-{\mu }^{-\nu })(1+o\left(1\right))$ as $x\to \infty$; expanding the logarithm gives (10), and assertion (11) follows on dividing by ${\epsilon }_L\left(x\right)\sim c_L{x}^{-\nu }$ as $x\to \infty$. □

Remark 1. 

Relation (10) shows that the shift remainder $w_{\mu }(*)$ does carry the decay rate $\mathcal{O}\left(x^{-\nu }\right)$ postulated in the earlier formulations, so the Basic Axioms in the present form are at least as strong as those of the earlier formulations discussed in Section 1. Relation (11) shows in what sense they are more precise. The factor $\nu /(1-{\mu }^{-\nu })$ depends on the shift and exceeds ν for every $\mu >1$, taking the value 1 only on the curve $\mu ={(1-\nu )}^{-1/\nu }$ of the parameter plane, and never at all when $\nu =1$. The index function and the shift remainder can therefore not be identified with one another: a constant computed by replacing one with the other differs from the true one by that factor. Working with ${\epsilon }_L(*)$ from the outset removes the difficulty, and with it the arbitrariness of the choice of μ.

We shall also use throughout the two auxiliary functions

$$\mathcal{V}\left(y\right):=y^{\nu }L(1/y)\mathrm{and}\mathsf{\Lambda }\left(x\right):={\displaystyle \frac{\ell \left(x\right)}{L\left(x\right)}}$$

(12)

for $y\in (0,1]$ and $x\ge 1$, together with the constant

$${\mathsf{\Lambda }}_{\infty }:=\underset{x\to \infty }{lim}\mathsf{\Lambda }\left(x\right)={\displaystyle \frac{{\ell }_{\infty }}{L_{\infty }}}\in (0,\infty ),$$

(13)

whose existence is guaranteed by Lemma 1. The constant ${\mathsf{\Lambda }}_{\infty }$ of (13) is the single number through which the immigration mechanism enters every limit theorem below. Hypothesis $\left[f_{\nu }\right]$ gives

$$\mathcal{V}\left(1\right)=L\left(1\right)=a_0.$$

(14)

Finally we record, for later use, the elementary consequence of Lemma 1 that

$${\mathsf{\Lambda }}_{\infty }-\mathsf{\Lambda }\left(x\right)={\mathsf{\Lambda }}_{\infty }[{\displaystyle \frac{c_{\ell }}{\delta x^{\delta }}}-{\displaystyle \frac{c_L}{\nu x^{\nu }}}]+o\left(x^{-min\{\nu ,\delta \}}\right)\mathrm{as}x\to \infty ,$$

(15)

which is obtained by writing

$${\mathsf{\Lambda }}_{\infty }-\mathsf{\Lambda }\left(x\right)={\displaystyle \frac{{\ell }_{\infty }(L\left(x\right)-L_{\infty })+L_{\infty }({\ell }_{\infty }-\ell \left(x\right))}{L_{\infty }L\left(x\right)}}$$

and inserting (8) and its analogue for $\ell (*)$. The bracket in (15) may vanish identically, as it does when $\nu =\delta$ and $c_{\ell }=c_L$, and in particular in the classical case $c_L=c_{\ell }=0$; this is why the remainder there is additive and not relative. If moreover $\left({\mathcal{A}}^{*}\right)$ and $\left({\mathcal{B}}^{*}\right)$ hold, then the same computation carried one step further sharpens the o-term of (15) to an explicit power,

$${\mathsf{\Lambda }}_{\infty }-\mathsf{\Lambda }\left(x\right)={\mathsf{\Lambda }}_{\infty }[{\displaystyle \frac{c_{\ell }}{\delta x^{\delta }}}-{\displaystyle \frac{c_L}{\nu x^{\nu }}}]+\mathcal{O}\left(x^{-min\{\nu ,\delta \}-{\rho }_{*}}\right)\mathrm{as}x\to \infty ,$$

(16)

with ${\rho }_{*}$ of (6): by Lemma 1 the two numerators carry the errors $\mathcal{O}\left(x^{-\nu -min\{\nu ,{\rho }_L\}}\right)$ and $\mathcal{O}\left(x^{-\delta -min\{\delta ,{\rho }_{\ell }\}}\right)$, and the expansion of $1/L\left(x\right)$ about $1/L_{\infty }$ contributes $\mathcal{O}\left(x^{-min\{\nu ,\delta \}-\nu }\right)$.

3. The Basic Lemma of the Critical Branching System

Throughout this section we work with the critical MBS $\left\{Z\right(t),t\in \mathcal{T}\}$ generated by the infinitesimal generating functions $f\left(s\right)$ alone, and with

$$R(t;s):=1-F(t;s)\mathrm{and}\tau (t;s):={\displaystyle \frac{1}{R(t;s)}},$$

together with the abbreviations

$$\tau \left(t\right):=\tau (t;0)\mathrm{and}q\left(t\right):=R(t;0).$$

The quantity $q\left(t\right)=\mathbb{P}\left\{Z\right(t)\in \mathcal{S}\}$ is the survival probability of the system started by a single ancestor. A statement describing the asymptotic behaviour of $\tau (t;s)$ is traditionally called the Basic Lemma of the theory of critical branching systems; under hypotheses of the present kind it goes back to Zolotarev [41] in continuous time and to Slack [37] and Seneta [34] in discrete time. We shall obtain it from an identity that is exact, and it is the exactness that makes the constants below accessible. We need the generating function of the invariant measure of the MBS,

$$M\left(s\right):={\int }_0^s{\displaystyle \frac{1}{f\left(x\right)}}\mathbf{d}x={\int }_1^{1/(1-s)}{\displaystyle \frac{1}{x^{1-\nu }L\left(x\right)}}\mathbf{d}x,$$

(17)

the second equality following from $\left[f_{\nu }\right]$ and the substitution $1-x=1/u$. Its coefficients are written $m_j$ throughout, so that $M\left(s\right)={\sum }_{j\in \mathcal{S}}{m}_j{s}^j$ for $s\in [0,1)$; that $M(*)$ is analytic on the whole unit disc, and hence that the expansion converges there, is part of Lemma 9 below, whose proof uses nothing from the intervening sections. Together with it we use the integral

$$\mathcal{I}\left(X\right):={\int }_1^X{\displaystyle \frac{x^{\nu }{L}^{\prime }\left(x\right)}{L^2\left(x\right)}}\mathbf{d}x,$$

(18)

which is finite for every $X\ge 1$ and nondecreasing when ${\epsilon }_L(*)\ge 0$.

Lemma 3. 

Let $\left[f_{\nu }\right]$ hold. Then for every $s\in [0,1)$ and every $t\in \mathcal{T}$

$${\displaystyle \frac{1}{\mathcal{V}\left(R\right(t;s\left)\right)}}=\nu [t+M\left(s\right)]+{\displaystyle \frac{1}{a_0}}-\mathcal{I}\left(\tau (t;s)\right).$$

(19)

Proof. 

Differentiating $Y^{\nu }/L\left(Y\right)$ and integrating the result over $[1,X]$ gives

$${\displaystyle \frac{X^{\nu }}{L\left(X\right)}}-{\displaystyle \frac{1}{L\left(1\right)}}=\nu {\int }_1^X{\displaystyle \frac{1}{Y^{1-\nu }L\left(Y\right)}}\mathbf{d}Y-\mathcal{I}\left(X\right),$$

valid for every $X\ge 1$. Take $X=\tau (t;s)$. The left-hand side is $1/\mathcal{V}\left(R(t;s)\right)-1/a_0$ by (12) and (14). In the integral on the right-hand side substitute $Y=1/(1-x)$ and use (17): it equals $M\left(F\right(t;s\left)\right)$, because $\tau (t;s)=1/(1-F(t;s\left)\right)$. Finally the backward equation (2) gives

$$\mathbf{d}M\left(F\right(t;s\left)\right)/\mathbf{d}t=f\left(F\right(t;s\left)\right)/f\left(F\right(t;s\left)\right)=1,$$

whence $M\left(F\right(t;s\left)\right)=M\left(s\right)+t$, and (19) follows. □

Identity (19) is exact; it involves no asymptotics and no hypothesis beyond $\left[f_{\nu }\right]$. Everything in this section is read off from it once the growth of $\mathcal{I}(*)$ is known. Put

$$\vartheta :={\displaystyle \frac{c_L}{L_{\infty }}}\mathrm{and}{\rho }_0:=min\{\nu ,{\rho }_L\}.$$

(20)

Lemma 4. 

Let $\left(\mathcal{A}\right)$ hold. Then

$$\mathcal{I}\left(X\right)=\vartheta lnX+o(lnX)\mathrm{as}X\to \infty .$$

(21)

If in addition $\left({\mathcal{A}}^{*}\right)$ holds, then the constant

$${\kappa }_L:={\int }_1^{\infty }[{\displaystyle \frac{x^{\nu }{L}^{\prime }\left(x\right)}{L^2\left(x\right)}}-{\displaystyle \frac{\vartheta }{x}}]\mathbf{d}x$$

(22)

is finite and

$$\mathcal{I}\left(X\right)=\vartheta lnX+{\kappa }_L+\mathcal{O}\left(X^{-{\rho }_0}\right)\mathrm{as}X\to \infty .$$

(23)

Proof. 

Recall the notation (20) and write the integrand of (18) as

$${\displaystyle \frac{x^{\nu }{L}^{\prime }\left(x\right)}{L^2\left(x\right)}}={\displaystyle \frac{1}{x}}·{\displaystyle \frac{x^{\nu }{\epsilon }_L\left(x\right)}{L\left(x\right)}},$$

which is legitimate because ${\epsilon }_L\left(x\right)=x{L}^{\prime }\left(x\right)/L\left(x\right)$. By $\left(\mathcal{A}\right)$ and Lemma 1 the second factor tends to $c_L/L_{\infty }=\vartheta$, and (21) follows on integrating. Under $\left({\mathcal{A}}^{*}\right)$ the deviation of that factor from its limit is

$${\displaystyle \frac{x^{\nu }{\epsilon }_L\left(x\right)}{L\left(x\right)}}-\vartheta ={\displaystyle \frac{x^{\nu }{\epsilon }_L\left(x\right)-c_L}{L\left(x\right)}}+c_L{\displaystyle \frac{L_{\infty }-L\left(x\right)}{L_{\infty }L\left(x\right)}}=\mathcal{O}\left(x^{-{\rho }_L}\right)+\mathcal{O}\left(x^{-\nu }\right),$$

by Lemma 1. Dividing by x produces an absolutely integrable function on $[1,\infty )$, so ${\kappa }_L$ is finite, and the tail of the integral is $\mathcal{O}\left(X^{-{\rho }_0}\right)$, which is (23). □

Lemma 4, and likewise every tail estimate of Section 6, Section 7 and Section 8, is the index-function counterpart of the general asymptotics of integrals of Karamata functions with a controlled remainder, for which see [7,12] and [6, §3.12]; we prove the versions we need inline, because what is wanted here is not the order of the tail but the exact constant in front of it, and that constant is $c_L$ or $c_{\ell }$ rather than the shift remainder. Combining Lemmas 3 and 4 gives the Basic Lemma in the form in which it will be used.

Theorem 1. 

Let $\left[f_{\nu }\right]$ and $\left(\mathcal{A}\right)$ hold. Then, uniformly in $s\in [0,d]$ for every $d<1$,

$${\displaystyle \frac{1}{\mathcal{V}\left(R\right(t;s\left)\right)}}=\nu [t+M\left(s\right)]+{\displaystyle \frac{1}{a_0}}-\vartheta ln\tau (t;s)+o(lnt)\mathrm{as}t\to \infty .$$

(24)

If in addition $\left({\mathcal{A}}^{*}\right)$ holds, then the remainder is exact:

$${\displaystyle \frac{1}{\mathcal{V}\left(R\right(t;s\left)\right)}}=\nu [t+M\left(s\right)]+{\displaystyle \frac{1}{a_0}}-{\kappa }_L-\vartheta ln\tau (t;s)+\mathcal{O}\left(t^{-{\rho }_0/\nu }\right)\mathrm{as}t\to \infty ,$$

(25)

uniformly in $s\in [0,d]$.

Proof. 

Insert (21) and (23) into (19). For the uniformity note that $R(t;s)\le R(t;0)=q\left(t\right)$ for $s\in [0,d]$ and $R(t;s)\ge R(t;d)$, so that

$$\tau \left(t\right)\le \tau (t;s)\le \tau (t;d)\mathrm{for}\mathrm{every}s\in [0,d]\mathrm{and}t\in \mathcal{T};$$

(26)

since $M\left(s\right)\le M\left(d\right)<\infty$ on $[0,d]$, identity (19) read together with (26) forces $\tau (t;s)\asymp \tau \left(t\right)$ uniformly on $[0,d]$, and in particular

$${\tau }^{-{\rho }_0}(t;s)=\mathcal{O}\left({\tau }^{-{\rho }_0}\left(t\right)\right)\mathrm{as}t\to \infty ,$$

(27)

uniformly on $[0,d]$. That $\tau \left(t\right)$ is of exact order $t^{1/\nu }$ is the content of (28) in Theorem 2 below, and with (27) it turns the remainder $\mathcal{O}\left({\tau }^{-{\rho }_0}(t;s)\right)$ of (23) into the uniform $\mathcal{O}\left(t^{-{\rho }_0/\nu }\right)$ of (25). The proof of Theorem 2 uses (19) at $s=0$ and (21) only, so that the order of proof is (19), (21), (28), (24) and there is no circularity. □

The choice $s=0$ in Theorem 1, for which $M\left(0\right)=0$ and $\tau (t;0)=\tau \left(t\right)$, yields the survival probability. The expansion below has three terms and no unspecified constants.

Theorem 2. 

Let $\left[f_{\nu }\right]$ and $\left(\mathcal{A}\right)$ hold. Then

$$q\left(t\right)={\left(\nu L_{\infty }t\right)}^{-1/\nu }(1+o\left(1\right))\mathrm{as}t\to \infty .$$

(28)

If in addition $\left({\mathcal{A}}^{*}\right)$ holds, then

$${\displaystyle \frac{1}{q^{\nu }\left(t\right)}}=\nu L_{\infty }t-{\displaystyle \frac{c_L}{\nu }}ln\left(\nu L_{\infty }t\right)+\mathfrak{c}+o\left(1\right)\mathrm{as}t\to \infty ,$$

(29)

where the constant, built from ${\kappa }_L$ of (22), is

$$\mathfrak{c}:=L_{\infty }[{\displaystyle \frac{1}{a_0}}-{\kappa }_L]-{\displaystyle \frac{c_L}{\nu }},$$

(30)

and equivalently

$$q\left(t\right)={\left(\nu L_{\infty }t\right)}^{-1/\nu }\{1+{\displaystyle \frac{c_{L}ln\left(\nu L_{\infty }t\right)-\nu \mathfrak{c}}{{\nu }^3{L}_{\infty }t}}+o(1/t)\}\mathrm{as}t\to \infty .$$

(31)

Proof. 

Put $s=0$ in (19). Since

$$1/\mathcal{V}\left(q\left(t\right)\right)={\tau }^{\nu }\left(t\right)/L\left(\tau \left(t\right)\right)$$

and $L\left(\tau \left(t\right)\right)\to L_{\infty }$ as $t\to \infty$, relation (21) applied to $\mathcal{I}\left(\tau \right(t\left)\right)$ gives ${\tau }^{\nu }\left(t\right)=L_{\infty }\nu t(1+o\left(1\right))$ as $t\to \infty$, which is (28). Under $\left({\mathcal{A}}^{*}\right)$ we use (23) instead and multiply by $L\left(\tau \right(t\left)\right)$:

$${\tau }^{\nu }\left(t\right)=L\left(\tau \left(t\right)\right)[\nu t+{\displaystyle \frac{1}{a_0}}-{\kappa }_L-\vartheta ln\tau \left(t\right)]+\mathcal{O}\left({\tau }^{-{\rho }_0}\left(t\right)\right).$$

By Lemma 1,

$$L\left(\tau \left(t\right)\right)=L_{\infty }-(L_{\infty }{c}_L/\nu ){\tau }^{-\nu }\left(t\right)+o\left({\tau }^{-\nu }\left(t\right)\right);$$

since ${\tau }^{-\nu }\left(t\right)\nu t\to 1/L_{\infty }$ as $t\to \infty$ by (28), the product of the first correction with $\nu t$ contributes exactly $-c_L/\nu$, that of the second is $o\left(1\right)$, and the product of both with the remaining bounded terms is $o\left(1\right)$. Because $L_{\infty }\vartheta =c_L$ and

$$ln\tau \left(t\right)={\displaystyle \frac{1}{\nu }}ln{\tau }^{\nu }\left(t\right)={\displaystyle \frac{1}{\nu }}ln\left(\nu L_{\infty }t\right)+o\left(1\right)\mathrm{as}t\to \infty$$

(32)

by (28), we arrive at

$${\tau }^{\nu }\left(t\right)=\nu L_{\infty }t-{\displaystyle \frac{c_L}{\nu }}ln\left(\nu L_{\infty }t\right)+L_{\infty }[{\displaystyle \frac{1}{a_0}}-{\kappa }_L]-{\displaystyle \frac{c_L}{\nu }}+o\left(1\right),$$

which is (29). Raising (29) to the power $-1/\nu$ gives (31). □

Remark 2. 

In [17,18] the leading term of (28) is written as $\mathcal{N}\left(t\right)/{\left(\nu t\right)}^{1/\nu }$, where $\mathcal{N}(*)\in {\mathcal{SV}}_{\infty }$ is defined implicitly by

$${\mathcal{N}}^{\nu }\left(t\right)L({\left(\nu t\right)}^{1/\nu }/\mathcal{N}\left(t\right))\to 1.$$

Under the Basic Axioms the auxiliary function is superfluous: by Lemma 1 the slowly varying factor $L(*)$ has a finite positive limit, so that $\mathcal{N}\left(t\right)\to 1/L_{\infty }^{1/\nu }$ as $t\to \infty$ and the normalisation is the explicit power ${\left(\nu L_{\infty }t\right)}^{-1/\nu }$. We therefore do not use $\mathcal{N}(*)$ anywhere below.

The comparison extends to the second term, and there the two accounts differ. Written in the notation of (20), the logarithmic correction of [17] is the one carried by (29) and (31) when $\vartheta =1$, whereas under the Basic Axioms ϑ is a free constant of the offspring generating function, equal to the limit of $x^{\nu }{\epsilon }_L\left(x\right)/L\left(x\right)$; Model T of Section 10 has $\vartheta =0.2$. The origin of the difference is the one recorded in Remark 1. The hypothesis there is imposed on the shift remainder $w_{\mu }(*)$ and reads $x^{\nu }{w}_{\mu }\left(x\right)/L\left(x\right)\to 1$ as $x\to \infty$; by (11) that hypothesis gives $\vartheta =\nu /(1-{\mu }^{-\nu })$, a quantity which still depends on the shift, and the value $\vartheta =1$ is what the further identification of $w_{\mu }(*)$ with ${\epsilon }_L(*)$ produces. The coefficient is open to inspection in a closed-form model, as Section 10 shows.

The comparison of $\tau (t;s)$ with $\tau \left(t\right)$ is needed in each of the sections that follow, and is recorded here once.

Lemma 5. 

Let $\left[f_{\nu }\right]$ hold. Then for every $s\in [0,1)$ and every $t\in \mathcal{T}$

$${\displaystyle \frac{1}{\mathcal{V}\left(R\right(t;s\left)\right)}}-{\displaystyle \frac{1}{\mathcal{V}\left(q\right(t\left)\right)}}=\nu M\left(s\right)-[\mathcal{I}\left(\tau (t;s)\right)-\mathcal{I}\left(\tau \left(t\right)\right)],$$

(33)

an identity in which the logarithmic terms of Theorem 1 cancel. If in addition $\left(\mathcal{A}\right)$ holds, then the bracket in (33) is $\mathcal{O}(1/t)$ and, uniformly in $s\in [0,d]$ for every $d<1$,

$${\left[\tau (t;s)\right]}^{\nu }-{\left[\tau \left(t\right)\right]}^{\nu }=\nu L_{\infty }M\left(s\right)+\mathcal{O}(1/t)\mathrm{as}t\to \infty$$

(34)

and

$$ln{\displaystyle \frac{\tau (t;s)}{\tau \left(t\right)}}={\displaystyle \frac{M\left(s\right)}{\nu t}}+o(1/t)\mathrm{as}t\to \infty .$$

(35)

In particular, uniformly in $s\in [0,d]$,

$$0\le \tau (t;s)-\tau \left(t\right)=\mathcal{O}\left({\tau }^{1-\nu }\left(t\right)\right)\mathrm{as}t\to \infty .$$

(36)

Proof. 

Subtracting (19) at $s=0$ from (19) at s gives (33). Let now $\left(\mathcal{A}\right)$ hold and fix $d<1$. The integrand of $\mathcal{I}(*)$ is $1/x$ times $x^{\nu }{\epsilon }_L\left(x\right)/L\left(x\right)$, and the latter is bounded by $\left(\mathcal{A}\right)$ and Lemma 1; hence, with a constant $C_0$ and uniformly on $[0,d]$,

$$|\mathcal{I}\left(\tau (t;s)\right)-\mathcal{I}\left(\tau \left(t\right)\right)|\le C_{0}ln{\displaystyle \frac{\tau (t;s)}{\tau \left(t\right)}}\le C_0{\displaystyle \frac{\tau (t;s)-\tau \left(t\right)}{\tau \left(t\right)}}.$$

(37)

Since $\tau (t;s)\asymp \tau \left(t\right)$ uniformly on $[0,d]$, as shown in the proof of Theorem 1, the bracket in (33) is bounded, so that ${\tau }^{\nu }(t;s)/L\left(\tau (t;s)\right)-{\tau }^{\nu }\left(t\right)/L\left(\tau \left(t\right)\right)$ is bounded too; as $L(*)$ has a finite positive limit, $\tau (t;s)/\tau \left(t\right)\to 1$ uniformly on $[0,d]$. Write $D:={\tau }^{\nu }(t;s)-{\tau }^{\nu }\left(t\right)\ge 0$ and multiply (33) by $L\left(\tau \right(t\left)\right)$; since $1/\mathcal{V}\left(R(t;s)\right)={\tau }^{\nu }(t;s)/L\left(\tau (t;s)\right)$ this reads

$$D=L\left(\tau \left(t\right)\right)[\nu M\left(s\right)-\mathcal{I}\left(\tau (t;s)\right)+\mathcal{I}\left(\tau \left(t\right)\right)]+{\tau }^{\nu }(t;s){\displaystyle \frac{L\left(\tau \right(t;s\left)\right)-L\left(\tau \right(t\left)\right)}{L\left(\tau \right(t;s\left)\right)}}.$$

The mean value theorem applied to $x\mapsto x^{1/\nu }$ gives $\tau (t;s)-\tau \left(t\right)\le D{\tau }^{1-\nu }(t;s)/\nu$, because $1/\nu -1$ is nonnegative. By (37) the bracket term is therefore $\mathcal{O}\left(D{\tau }^{-\nu }\left(t\right)\right)$, and by (9) so is the last term, since

$$\mathcal{O}\left({\tau }^{\nu }(t;s)D{\tau }^{1-\nu }(t;s){\tau }^{-1-\nu }\left(t\right)\right)=\mathcal{O}\left(D{\tau }^{-\nu }\left(t\right)\right),$$

both uniformly on $[0,d]$. Both are $o\left(D\right)$, so that $D=\mathcal{O}\left(1\right)$, and the bracket is of the order ${\tau }^{-\nu }\left(t\right)$, that is $\mathcal{O}(1/t)$ by (28). Solving for D now gives (34), the replacement of $L\left(\tau \right(t\left)\right)$ by $L_{\infty }$ costing $\mathcal{O}\left({\tau }^{-\nu }\left(t\right)\right)=\mathcal{O}(1/t)$ by Lemma 1. Finally, by (34) and Theorem 2,

$$ln{\displaystyle \frac{\tau (t;s)}{\tau \left(t\right)}}={\displaystyle \frac{1}{\nu }}ln[1+{\displaystyle \frac{\nu L_{\infty }M\left(s\right)}{{\tau }^{\nu }\left(t\right)}}+\mathcal{O}(1/t^2)]={\displaystyle \frac{M\left(s\right)}{\nu t}}+o(1/t),$$

which is (35). Finally (36) is the mean value bound above read with $D=\mathcal{O}\left(1\right)$, obtained above, and $\tau (t;s)\asymp \tau \left(t\right)$ uniformly on $[0,d]$ by (26). □

We turn to the derivative of $R(t;s)$, which carries the local probabilities. Here too an exact identity is available.

Lemma 6. 

Let $\left[f_{\nu }\right]$ hold. Then for every $s\in [0,1)$ and $t\in \mathcal{T}$

$${\displaystyle \frac{\partial R(t;s)}{\partial s}}=-{\displaystyle \frac{R(t;s)}{f\left(s\right)}}\mathcal{V}\left(R(t;s)\right)\mathrm{and}\mathrm{in}\mathrm{particular}{\mathsf{p}}_1\left(t\right)={\displaystyle \frac{q\left(t\right)\mathcal{V}\left(q\right(t\left)\right)}{a_0}}.$$

(38)

Proof. 

Equating the right-hand sides of (2) and (3) gives $\partial F(t;s)/\partial s=f\left(F\right(t;s\left)\right)/f\left(s\right)$, so that $\partial R/\partial s=-f(1-R)/f\left(s\right)$; and $f(1-R)=R·\mathcal{V}\left(R\right)$ by $\left[f_{\nu }\right]$ and (12). Setting $s=0$ and using $\partial R(t;s)/\partial s{|}_{s=0}=-{\mathsf{p}}_1\left(t\right)$ together with $f\left(0\right)=a_0$ yields the second assertion. □

Theorem 3. 

Let $\left[f_{\nu }\right]$ and $\left({\mathcal{A}}^{*}\right)$ hold, and put $\mathfrak{d}:=1/a_0-{\kappa }_L$. Then

$${\displaystyle \frac{{\mathsf{p}}_1\left(t\right)}{q\left(t\right)}}={\displaystyle \frac{1}{a_0\nu t}}\{1+{\displaystyle \frac{\vartheta ln\left(\nu L_{\infty }t\right)-\nu \mathfrak{d}}{{\nu }^{2}t}}+o(1/t)\}\mathrm{as}t\to \infty ,$$

(39)

and consequently

$$t^{1+1/\nu }{\mathsf{p}}_1\left(t\right)⟶{\displaystyle \frac{L_{\infty }}{a_0{\left(\nu L_{\infty }\right)}^{1+1/\nu }}}\mathrm{as}t\to \infty .$$

(40)

Proof. 

By (38) the ratio ${\mathsf{p}}_1\left(t\right)/q\left(t\right)$ equals $\mathcal{V}\left(q\left(t\right)\right)/a_0$, and $\mathcal{V}\left(q\right(t\left)\right)$ is the reciprocal of the left-hand side of (25), so that

$${\displaystyle \frac{{\mathsf{p}}_1\left(t\right)}{q\left(t\right)}}={\displaystyle \frac{\mathcal{V}\left(q\right(t\left)\right)}{a_0}}={\displaystyle \frac{1}{a_0[\nu t+\mathfrak{d}-\vartheta ln\tau \left(t\right)+o\left(1\right)]}}$$

by (25) with $s=0$. Expanding the bracket and using (32) gives (39). For (40) write ${\mathsf{p}}_1\left(t\right)=q^{1+\nu }\left(t\right)L\left(\tau \left(t\right)\right)/a_0$, again by (38) and (12), and insert (28). □

Finally we record the generating function $M\left(s\right)$ itself. Setting $t=0$ in Lemma 3 gives an identity that is worth isolating, because the form of $M\left(s\right)$ enters the conditional limit theorems of Section 4.

Corollary 1. 

Let $\left[f_{\nu }\right]$ hold. Then for every $s\in [0,1)$

$$\nu M\left(s\right)={\displaystyle \frac{1}{\mathcal{V}(1-s)}}-{\displaystyle \frac{1}{a_0}}+\mathcal{I}\left({\displaystyle \frac{1}{1-s}}\right).$$

(41)

Remark 3. 

The term $\mathcal{I}(1/(1-s\left)\right)$ in (41) cannot be dispensed with. It vanishes identically if and only if $L(*)$ is constant, which by $\left[f_{\nu }\right]$ and (14) means $f\left(s\right)=a_0{(1-s)}^{1+\nu }$; in the finite variance case $\nu =1$ one recovers the familiar $M\left(s\right)=s/\left(a_0(1-s)\right)$. For a genuinely varying $L(*)$ the reduced expression $[1/\mathcal{V}(1-s)-1/a_0]/\nu$, obtained by suppressing that term, differs from $M\left(s\right)$ by a factor of the order of unity rather than by a quantity of smaller order; the ratio of the two is given for Model T in Section 10. Since the two agree identically when $L(*)$ is constant, the distinction is invisible in the finite variance theory and becomes visible only in the present generality.

It is also invisible in the limit $s\uparrow 1$. Under $\left(\mathcal{A}\right)$ the omitted term is $\mathcal{I}\left(X\right)=\vartheta lnX+o(lnX)$ by (21), while $1/\mathcal{V}(1-s)$ grows like a power of $1/(1-s)$; the reduced expression is therefore the correct leading term of $M\left(s\right)$ as $s\uparrow 1$, and the ratio of the two tends to unity, as Figure 3 shows for Model T. This is why the omission leaves untouched every statement in which $M(*)$ enters asymptotically or is carried symbolically, and shows itself only when the formula is read as an identity at a fixed s. It is so read in [17, Thm. 7], where the term $\mathcal{I}(1/(1-s\left)\right)$ is absent from the displayed expression for $M\left(s\right)$; (41) is the corrected form, and it is the one used throughout the present paper.

4. The Conditioned Branching System and Local Ratios

We stay with the MBS without immigration for one more section and record two consequences of Lemma 3 that are of interest in their own right and that sharpen the classical conditional limit theorems of [23,32,33,37,38]. The limits themselves are known in the present generality; what is taken over and what is added is set out in Section 1, and we do not repeat it. What is added here is the second-order correction of Theorem 4 below, quadratic in $M\left(s\right)$, together with the uniformity in s and the transfer to the individual probabilities. Set

$${\mathsf{p}}_j^{\mathcal{S}}\left(t\right):=\mathbb{P}\{Z\left(t\right)=j|Z\left(t\right)\in \mathcal{S}\}={\displaystyle \frac{{\mathsf{p}}_j\left(t\right)}{q\left(t\right)}}\mathrm{and}{F}^{\mathcal{S}}(t;s):=\sum _{j\in \mathcal{S}}{\mathsf{p}}_j^{\mathcal{S}}\left(t\right)s^j=1-{\displaystyle \frac{R(t;s)}{q\left(t\right)}},$$

the last equality being immediate from ${\sum }_{j\in \mathcal{S}}{\mathsf{p}}_j\left(t\right)s^j=F(t;s)-{\mathsf{p}}_0\left(t\right)$. It is convenient to abbreviate

$$A\left(t\right):={\displaystyle \frac{1}{\mathcal{V}\left(q\right(t\left)\right)}}=\nu t+\mathfrak{d}-\vartheta ln\tau \left(t\right)+o\left(1\right)\mathrm{as}t\to \infty ,$$

(42)

the second equality being (25) at $s=0$ under $\left({\mathcal{A}}^{*}\right)$.

Theorem 4. 

Let $\left[f_{\nu }\right]$ and $\left(\mathcal{A}\right)$ hold. Then, uniformly in $s\in [0,d]$ for every $d<1$,

$$A\left(t\right)F^{\mathcal{S}}(t;s)=M\left(s\right)+\mathcal{O}(1/t)\mathrm{as}t\to \infty ,$$

(43)

and the deviation in (43) is itself computable to one order more,

$$F^{\mathcal{S}}(t;s)={\displaystyle \frac{M\left(s\right)}{A\left(t\right)}}-{\displaystyle \frac{(1+\nu )M^2\left(s\right)}{2A^2\left(t\right)}}+o(1/t^2)\mathrm{as}t\to \infty ,$$

(44)

again uniformly in $s\in [0,d]$. If in addition $\left({\mathcal{A}}^{*}\right)$ holds, so that $A\left(t\right)$ may be replaced by its expansion (42), then

$$\nu t{F}^{\mathcal{S}}(t;s)=M\left(s\right)\{1+{\displaystyle \frac{\vartheta ln\left(\nu L_{\infty }t\right)-\nu \mathfrak{d}}{{\nu }^{2}t}}-{\displaystyle \frac{(1+\nu )M\left(s\right)}{2\nu t}}+o(1/t)\}\mathrm{as}t\to \infty ,$$

(45)

uniformly in $s\in [0,d]$, the last term being the only part of the correction that carries a dependence on s.

Proof. 

Write $A_s\left(t\right):=1/\mathcal{V}\left(R(t;s)\right)$. By (12),

$${\left({\displaystyle \frac{R(t;s)}{q\left(t\right)}}\right)}^{\nu }={\displaystyle \frac{\mathcal{V}\left(R\right(t;s\left)\right)}{\mathcal{V}\left(q\right(t\left)\right)}}·{\displaystyle \frac{L\left(\tau \right(t\left)\right)}{L\left(\tau \right(t;s\left)\right)}}={\displaystyle \frac{A\left(t\right)}{A_s\left(t\right)}}·{\displaystyle \frac{L\left(\tau \right(t\left)\right)}{L\left(\tau \right(t;s\left)\right)}}.$$

(46)

Identity (33) gives $A_s\left(t\right)=A\left(t\right)+\nu M\left(s\right)+\mathcal{O}(1/t)$ as $t\to \infty$. The second factor of (46) is not merely bounded: by (9) and the bound (36) of Lemma 5 it satisfies

$${\displaystyle \frac{L\left(\tau \right(t;s\left)\right)}{L\left(\tau \right(t\left)\right)}}=1+\mathcal{O}(1/t^2)\mathrm{as}t\to \infty ,$$

uniformly on $[0,d]$, the exact form of that factor being (48) below. Hence, as $t\to \infty$, one has $R(t;s)/q\left(t\right)=1-M\left(s\right)/A\left(t\right)+\mathcal{O}(1/t^2)$, which is (43).

For (44) the two factors of (46) are needed one order further. In (33) the bracket is

$$\mathcal{I}\left(\tau (t;s)\right)-\mathcal{I}\left(\tau \left(t\right)\right)={\int }_{\tau \left(t\right)}^{\tau (t;s)}{\mathcal{I}}^{\prime }\left(x\right)\mathbf{d}x\mathrm{with}{\mathcal{I}}^{\prime }\left(x\right)={\displaystyle \frac{x^{\nu }{L}^{\prime }\left(x\right)}{L^2\left(x\right)}}={\displaystyle \frac{\vartheta }{x}}+o\left({\displaystyle \frac{1}{x}}\right)$$

by $\left(\mathcal{A}\right)$, so that by (35) the bracket equals $\vartheta M\left(s\right)/\left(\nu t\right)+o(1/t)$. Consequently

$$A_s\left(t\right)-A\left(t\right)=\nu M\left(s\right)-{\displaystyle \frac{\vartheta M\left(s\right)}{\nu t}}+o(1/t)\mathrm{as}t\to \infty .$$

(47)

For the second factor of (46), $L^{\prime }\left(x\right)=L\left(x\right){\epsilon }_L\left(x\right)/x=L_{\infty }{c}_L{x}^{-1-\nu }+o\left(x^{-1-\nu }\right)$ as $x\to \infty$, so that integration between the two arguments, using (34) and $\vartheta =c_L/L_{\infty }$, gives

$${\displaystyle \frac{L\left(\tau \right(t\left)\right)}{L\left(\tau \right(t;s\left)\right)}}=1-{\displaystyle \frac{\vartheta M\left(s\right)}{{\nu }^2{t}^2}}+o(1/t^2)\mathrm{as}t\to \infty .$$

(48)

Write $y:=[A_s\left(t\right)-A\left(t\right)]/A\left(t\right)$ with (47), and expand the product of the two factors of (46) to order $t^{-2}$ with (48). Taking the $\nu$-th root by

$${(1-\varpi )}^{1/\nu }=1-{\displaystyle \frac{\varpi }{\nu }}+{\displaystyle \frac{(1-\nu ){\varpi }^2}{2{\nu }^2}}+\mathcal{O}\left({\varpi }^3\right)\mathrm{with}\varpi =y-y^2+{\displaystyle \frac{\vartheta M\left(s\right)}{{\nu }^2{t}^2}}+o\left({\displaystyle \frac{1}{t^2}}\right),$$

the two terms in $\vartheta M\left(s\right)$ cancel, the two terms in $y^2$ combine with the coefficients $-1/\nu$ and $-(1-\nu )/\left(2{\nu }^2\right)$, and (44) follows, uniformly on $[0,d]$. Nothing beyond $\left(\mathcal{A}\right)$ has been used to this point, $A\left(t\right)$ having been carried unexpanded. It remains, under $\left({\mathcal{A}}^{*}\right)$, to multiply (44) by $\nu t/M\left(s\right)$ and to insert (42) together with (32), which gives (45). □

Remark 4. 

The unnormalised form (44) is the sharper of the two, and it states the content of the theorem without constants: the deviation of $A\left(t\right)F^{\mathcal{S}}(t;s)$ from $M\left(s\right)$ is asymptotically $-(1+\nu )M^2\left(s\right)/\left(2A\left(t\right)\right)$, quadratic in $M\left(s\right)$ and negative. In the normalised form (45) the correction splits into a part free of s, namely the second-order term of the survival probability transported through $A\left(t\right)$, and the term $-(1+\nu )M\left(s\right)/\left(2\nu t\right)$, which is not. The latter is the first place at which the conditioned law is seen to approach its limit at a rate depending on the argument, and it is governed by $M(*)$ alone. It does not disturb the ratio limit: dividing (45) at $s_1$ by the same relation at $s_2$ leaves $M\left(s_1\right)/M\left(s_2\right)$ with a relative error of order $1/t$, so that the limit of the ratio is unaffected.

The corresponding statement for the ratios to ${\mathsf{p}}_1\left(t\right)$ is even simpler, because of the exact formula (38). Let

$$V(t;s):=\sum _{j\in \mathcal{S}}{\displaystyle \frac{{\mathsf{p}}_j\left(t\right)}{{\mathsf{p}}_1\left(t\right)}}{s}^j.$$

Theorem 5. 

Let $\left[f_{\nu }\right]$ and $\left(\mathcal{A}\right)$ hold. Then, uniformly in $s\in [0,d]$ for every $d<1$,

$$V(t;s)=a_{0}M\left(s\right)+\mathcal{O}(1/t)\mathrm{as}t\to \infty ,$$

(49)

so that $V(t;s)\to V\left(s\right):=a_{0}M\left(s\right)={\sum }_{j\in \mathcal{S}}{v}_j{s}^j$ as $t\to \infty$, with $v_j=a_0{m}_j$. The $v_j$ are nonnegative, being by Theorem 12 below the limits of the nonnegative ratios ${\mathsf{p}}_j\left(t\right)/{\mathsf{p}}_1\left(t\right)$; that theorem uses of the present one only (49), so the reference is not circular.

Proof. 

By definition

$$V(t;s)=q\left(t\right)F^{\mathcal{S}}(t;s)/{\mathsf{p}}_1\left(t\right),$$

and by (38)

$$q\left(t\right)/{\mathsf{p}}_1\left(t\right)=a_0/\mathcal{V}\left(q\left(t\right)\right)=a_{0}A\left(t\right).$$

Hence $V(t;s)=a_{0}A\left(t\right)F^{\mathcal{S}}(t;s)$ and (49) is (43). □

Read together with (40) of Theorem 3, the limit (49) is a local limit theorem for the branching system without immigration: writing ${\mathsf{p}}_j\left(t\right)={\mathsf{p}}_1\left(t\right)[{\mathsf{p}}_j\left(t\right)/{\mathsf{p}}_1\left(t\right)]$ one obtains

$$t^{1+1/\nu }{\mathsf{p}}_j\left(t\right)⟶{\displaystyle \frac{L_{\infty }{v}_j}{a_0{\left(\nu L_{\infty }\right)}^{1+1/\nu }}}\mathrm{as}t\to \infty$$

(50)

for each $j\in \mathcal{S}$, the passage to the coefficients being legitimate by Theorem 12 below, which supplies it with a rate.

Remark 5. 

One point about the conditioned system should be recorded. The conditioned probabilities ${\mathsf{p}}_j^{\mathcal{S}}\left(t\right)$ are obtained by conditioning on the event $\left\{Z\right(t)\in \mathcal{S}\}$ separately for each $t\in \mathcal{T}$, and this operation does not produce a time-homogeneous Markov chain: the conditioning event changes with t, so the family $\left\{{\mathsf{p}}_j^{\mathcal{S}}\left(t\right)\right\}$ does not satisfy the Kolmogorov–Chapman equation. The Q-process, which conditions on non-extinction in the remote future, is a genuine Markov chain precisely because its conditioning event does not depend on t; see [24,32]. The statements (43) to (49), and those of Section 9, concern the conditioned distributions alone and use no Markov property of them, so nothing below depends on the point; we make it only to explain why no such property is invoked.

5. Classification of the State Space

We now return to the system with immigration. Everything rests on the following change of variable, which converts the time integral in (1) into an integral over the state variable and is used again in each of the three sections that follow.

Lemma 7. 

Let $\left[f_{\nu }\right]$ and $\left[h_{\delta }\right]$ hold. Then for every $s\in [0,1)$ and $t\in \mathcal{T}$

$$ln\mathcal{P}(t;s)=-{\int }_{1/(1-s)}^{\tau (t;s)}{\displaystyle \frac{\mathsf{\Lambda }\left(x\right)}{x^{1+\gamma }}}\mathbf{d}x.$$

(51)

Proof. 

By the backward equation (2) the substitution $u=F(v;s)$ turns ${\int }_0^{t}h\left(F(v;s)\right)\mathbf{d}v$ into ${\int }_s^{F(t;s)}[h\left(u\right)/f\left(u\right)]\mathbf{d}u$. By $\left[f_{\nu }\right]$, $\left[h_{\delta }\right]$ and (5),

$${\displaystyle \frac{h\left(u\right)}{f\left(u\right)}}=-{\displaystyle \frac{\ell (1/(1-u\left)\right)}{L(1/(1-u\left)\right)}}{\displaystyle \frac{1}{{(1-u)}^{1+\nu -\delta }}}=-{\displaystyle \frac{\mathsf{\Lambda }(1/(1-u\left)\right)}{{(1-u)}^{1-\gamma }}},$$

and the further substitution $1-u=1/x$, which carries $u=F(t;s)$ to $x=\tau (t;s)$, gives (51). □

The three regimes of (51) are governed by the sign of $\gamma$, because the integrand is asymptotically ${\mathsf{\Lambda }}_{\infty }{x}^{-1-\gamma }$: the integral diverges like a power of $\tau (t;s)$ for $\gamma <0$, like a logarithm for $\gamma =0$, and converges for $\gamma >0$. This is the whole of the classification, and the threshold at the boundary is the coefficient of the logarithm.

The criterion that is being evaluated is classical, and it is worth writing down in order to see what the theorem below adds to it. The chain is irreducible and honest by Section 2, so that $\mathcal{E}$ is recurrent if and only if ${\int }_0^{\infty }{p}_{00}\left(t\right)\mathbf{d}t=\infty$ and is positive recurrent if and only if $p_{00}\left(t\right)$ has a positive limit as $t\to \infty$; see [2]. Since $p_{00}\left(t\right)$ is (1) at $i=s=0$ and the substitution $v=F(t;0)$, for which $\mathbf{d}v=f\left(v\right)\mathbf{d}t$ by (2), carries $t\in \mathcal{T}$ onto $v\in [0,1)$, the first of the two integrals is

$${\int }_0^{\infty }{p}_{00}\left(t\right)\mathbf{d}t={\int }_0^1{\displaystyle \frac{1}{f\left(v\right)}}exp\left\{{\int }_0^v{\displaystyle \frac{h\left(u\right)}{f\left(u\right)}}\mathbf{d}u\right\}\mathbf{d}v.$$

(52)

In this form the criterion goes back to Pakes [30] and is developed in Chen and Renshaw [8] and in Li, Chen and Pakes [25]; the dichotomy between $\gamma >0$ and $\gamma <0$ is what reading it under $\left[f_{\nu }\right]$ and $\left[h_{\delta }\right]$ gives at once, and is recorded in [18,25]. The substance of the theorem below is therefore the boundary $\gamma =0$, where the sign carries no information and the two sides of (52) have to be separated by a second number; we state the three strata together because the classification is only complete when the boundary is included, and prove them from (51) at $s=0$ rather than from (52), the exponent of (51) being available to us in closed form.

Theorem 6. 

Let the Basic Axioms hold. Then the state space $\mathcal{E}$ of the MBIS is

(i)

positive recurrent if $\gamma >0$;

(ii)

null recurrent if $\gamma =0$ and ${\mathsf{\Lambda }}_{\infty }\le \nu$, and transient if $\gamma =0$ and ${\mathsf{\Lambda }}_{\infty }>\nu$;

(iii)

transient if $\gamma <0$.

Proof. 

We use the two criteria recalled above. By (51) at $s=0$,

$$ln{p}_{00}\left(t\right)=-{\int }_1^{\tau \left(t\right)}{\displaystyle \frac{\mathsf{\Lambda }\left(x\right)}{x^{1+\gamma }}}\mathbf{d}x,$$

(53)

and by Theorem 2, $\tau \left(t\right)={\left(\nu L_{\infty }t\right)}^{1/\nu }(1+o\left(1\right))$ as $t\to \infty$.

Let $\gamma >0$. Since $\mathsf{\Lambda }\left(x\right)\to {\mathsf{\Lambda }}_{\infty }$ and ${\int }_1^{\infty }{x}^{-1-\gamma }\mathbf{d}x<\infty$, the integral in (53) converges as $t\to \infty$, so that $p_{00}\left(t\right)$ tends to a strictly positive limit and (i) holds.

Let $\gamma =0$. Then (53) reads

$$ln{p}_{00}\left(t\right)=-{\mathsf{\Lambda }}_{\infty }ln\tau \left(t\right)-{\int }_1^{\tau \left(t\right)}{\displaystyle \frac{\mathsf{\Lambda }\left(x\right)-{\mathsf{\Lambda }}_{\infty }}{x}}\mathbf{d}x,$$

and by (15) with $\delta =\nu$ the last integrand is $\mathcal{O}\left(x^{-1-\nu }\right)$, so that the last integral converges to a finite limit. Hence

$$p_{00}\left(t\right)={\left(\nu L_{\infty }t\right)}^{-{\mathsf{\Lambda }}_{\infty }/\nu }(C+o\left(1\right))\mathrm{as}t\to \infty$$

(54)

for a constant $C>0$, and ${\int }^{\infty }{p}_{00}\left(t\right)\mathbf{d}t$ diverges precisely when ${\mathsf{\Lambda }}_{\infty }/\nu \le 1$. Since $p_{00}\left(t\right)\to 0$ as $t\to \infty$ in either case, recurrence is null, and (ii) holds.

Let $\gamma <0$. Then $\left|\gamma \right|=-\gamma >0$ and the integrand of (53) is at least $\frac{1}{2}{\mathsf{\Lambda }}_{\infty }{x}^{\left|\gamma \right|-1}$ for all large x, so that

$$ln{p}_{00}\left(t\right)\le -({\mathsf{\Lambda }}_{\infty }/\left(3\left|\gamma \right|\right)){\tau }^{\left|\gamma \right|}\left(t\right)$$

eventually. As ${\tau }^{\left|\gamma \right|}\left(t\right)$ is of exact order $t^{\left|\gamma \right|/\nu }$ with $\left|\gamma \right|/\nu >0$, the function $p_{00}\left(t\right)$ decays at a stretched exponential rate and is integrable, so (iii) holds. □

Remark 6. 

Part (iii) and the positive recurrence in part (i) recover the dichotomy of [18,25], itself the reading of (52) under the hypotheses $\left[f_{\nu }\right]$ and $\left[h_{\delta }\right]$; part (ii) is new and closes the boundary that the sign criterion leaves open. In [17] that stratum was set aside on the ground that $\gamma =0$ forces $h\left(s\right)=f^{\prime }\left(s\right)$ and hence a different model, the Markov Q-process. The implication does not hold: $\gamma =0$ says only that $\delta =\nu$, and it leaves ${\mathsf{\Lambda }}_{\infty }$ free, which is precisely why the stratum splits. It also reduces to the classical statement. Indeed, in the finite variance case $\nu =\delta =1$ with $L\left(x\right)\to L_{\infty }$ and $\ell \left(x\right)\to h^{\prime }(1-)$ as $x\to \infty$ we have $\gamma =0$ and ${\mathsf{\Lambda }}_{\infty }=h^{\prime }(1-)/L_{\infty }=\lambda$, where λ is exactly the parameter of Sevastyanov [35] and Pakes [30], and the threshold ${\mathsf{\Lambda }}_{\infty }\le \nu$ becomes $\lambda \le 1$. Thus the classical criterion is the section of the boundary stratum at $\nu =1$. Note also that the trichotomy is genuinely two-dimensional: on the boundary $\gamma =0$ the sign of γ carries no information at all and the whole of the classification is done by ${\mathsf{\Lambda }}_{\infty }$.

6. The Transient Regime $\gamma <0$

Throughout this section $\gamma <0$, so that $\left|\gamma \right|=\nu -\delta$ lies in $(0,\nu )$, and we assume $\chi =2\delta -\nu >0$. By Theorem 6 the state space is transient and the transition functions decay at a stretched exponential rate. The correct normalisation is

$$\mathsf{\Theta }\left(t\right):={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\left|\gamma \right|}}{\left[\tau \left(t\right)\right]}^{\left|\gamma \right|},$$

(55)

and we introduce the three quantities

$$\mathcal{K}\left(s\right):={\int }_{1/(1-s)}^{\infty }{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }-\mathsf{\Lambda }\left(u\right)}{u^{1-\left|\gamma \right|}}}\mathbf{d}u\mathrm{and}\mathcal{J}(t;s):={\int }_{\tau (t;s)}^{\infty }{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }-\mathsf{\Lambda }\left(u\right)}{u^{1-\left|\gamma \right|}}}\mathbf{d}u$$

(56)

and

$$\mathcal{D}(t;s):={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\left|\gamma \right|}}({\left[\tau \left(t\right)\right]}^{\left|\gamma \right|}-{\left[\tau (t;s)\right]}^{\left|\gamma \right|}).$$

(57)

Both integrals in (56) converge. Indeed, by (15) the integrand is

$${\mathsf{\Lambda }}_{\infty }[c_{\ell }/\delta ]u^{-1-\chi }+\mathcal{O}\left(u^{-1-\delta }\right)+o\left(u^{-1-\chi }\right),$$

because $\delta -\left|\gamma \right|=\chi$ and $\nu -\left|\gamma \right|=\delta$; it is therefore $\mathcal{O}\left(u^{-1-\chi }\right)$ as $u\to \infty$, and $\chi >0$ suffices for convergence; and when $c_{\ell }>0$ the integrand is of exactly that order, so that $\chi >0$ is then also necessary. This is where the hypothesis $\chi >0$ enters, and, as long as $c_{\ell }>0$, it is the only place.

The hypotheses $\left(\mathcal{A}\right)$ and $\left(\mathcal{B}\right)$ are needed already in the identity below, since ${\mathsf{\Lambda }}_{\infty }$ enters the definitions (55)–(57) and the convergence of the two integrals (56) was read off from (15).

Lemma 8. 

Let the Basic Axioms hold with $\gamma <0$ and $\chi >0$. Then for every $s\in [0,1)$ and $t\in \mathcal{T}$

$$e^{\mathsf{\Theta }\left(t\right)}\mathcal{P}(t;s)=\mathcal{U}\left(s\right)exp\{\mathcal{D}(t;s)-\mathcal{J}(t;s)\},$$

(58)

where

$$\mathcal{U}\left(s\right):=exp\{{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\left|\gamma \right|}}{\displaystyle \frac{1}{{(1-s)}^{\left|\gamma \right|}}}+\mathcal{K}\left(s\right)\}.$$

(59)

Proof. 

Split the integrand of (51), in which $1+\gamma =1-\left|\gamma \right|$, into ${\mathsf{\Lambda }}_{\infty }{u}^{\left|\gamma \right|-1}$ and

$$-[{\mathsf{\Lambda }}_{\infty }-\mathsf{\Lambda }\left(u\right)]u^{\left|\gamma \right|-1}.$$

The first part integrates to

$$({\mathsf{\Lambda }}_{\infty }/\left|\gamma \right|)[{\tau }^{\left|\gamma \right|}(t;s)-{(1-s)}^{-\left|\gamma \right|}]$$

and the second to $\mathcal{J}(t;s)-\mathcal{K}\left(s\right)$. Adding $\mathsf{\Theta }\left(t\right)$ and using (57) gives (58). □

Identity (58) is exact. The theorem below merely evaluates the two terms in its exponent. Note that (58) requires no relation between ${\mathsf{\Lambda }}_{\infty }$ and $\left|\gamma \right|$. The formulation of [17], which normalises by $exp\left\{{\tau }^{\left|\gamma \right|}\left(t\right)\right\}$ instead of by $e^{\mathsf{\Theta }\left(t\right)}$, is the special case ${\mathsf{\Lambda }}_{\infty }=\left|\gamma \right|$ of (55). Under that same hypothesis [16, Thm. 2.2] bounds the relative deviation by $\mathcal{O}(\ell \left(\tau \left(t\right)\right){\tau }^{-\chi }\left(t\right))$ as $t\to \infty$; what Theorem 7 below adds is the constant in front of that order, the freedom in ${\mathsf{\Lambda }}_{\infty }$, and the uniformity in s.

Theorem 7. 

Let the Basic Axioms hold with $\gamma <0$, $\chi >0$ and $c_{\ell }>0$. Then, uniformly in $s\in [0,d]$ for every $d<1$,

$$e^{\mathsf{\Theta }\left(t\right)}\mathcal{P}(t;s)=\mathcal{U}\left(s\right)(1-\zeta (t;s))\mathrm{for}\mathrm{every}t\in \mathcal{T},$$

(60)

where the relative deviation $\zeta (t;s)$ tends to zero as $t\to \infty$, is free of s to leading order, and equals

$$\zeta (t;s)={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }{c}_{\ell }}{\delta \chi }}{\displaystyle \frac{1}{{\left(\nu L_{\infty }t\right)}^{\chi /\nu }}}(1+o\left(1\right))\mathrm{as}t\to \infty .$$

(61)

If in addition $\left({\mathcal{A}}^{*}\right)$ and $\left({\mathcal{B}}^{*}\right)$ hold, then the $o\left(1\right)$ of (61) is exact:

$$\zeta (t;s)={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }{c}_{\ell }}{\delta \chi }}{\displaystyle \frac{1}{{\left(\nu L_{\infty }t\right)}^{\chi /\nu }}}(1+\mathcal{O}\left(t^{-{\rho }_1/\nu }\right))\mathrm{as}t\to \infty ,$$

(62)

where ${\rho }_1:=min\{{\rho }_{*},\left|\gamma \right|,\chi \}$ and ${\rho }_{*}$ is the rate (6). The coefficients of the power series expansion $\mathcal{U}\left(s\right)={\sum }_{j\in \mathcal{E}}{u}_j{s}^j$ are nonnegative and generate an invariant measure for the transition functions, that is,

$$u_j=\sum _{i\in \mathcal{E}}{u}_i{p}_{ij}\left(t\right)\mathrm{for}\mathrm{all}j\in \mathcal{E},t\in \mathcal{T}.$$

(63)

Proof. 

We evaluate the two terms in the exponent of (58).

By (15) and $\delta -\left|\gamma \right|=\chi$, $\nu -\left|\gamma \right|=\delta$, the integrand of $\mathcal{J}(t;s)$ equals

$${\mathsf{\Lambda }}_{\infty }[c_{\ell }/\delta ]u^{-1-\chi }-{\mathsf{\Lambda }}_{\infty }[c_L/\nu ]u^{-1-\delta }+o\left(u^{-1-\chi }\right),$$

so that

$$\mathcal{J}(t;s)={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }{c}_{\ell }}{\delta \chi }}{\displaystyle \frac{1}{{\tau }^{\chi }(t;s)}}(1+o\left(1\right))\mathrm{as}t\to \infty ,$$

(64)

the second contribution being of the smaller order ${\tau }^{-\delta }(t;s)$ because $\chi <\delta$ is equivalent to $\delta <\nu$. Here $c_{\ell }>0$ is used. By Theorem 2 and the uniform two-sided bound (26) we may replace $\tau (t;s)$ in (64) by ${\left(\nu L_{\infty }t\right)}^{1/\nu }$, which gives (61).

Under $\left({\mathcal{A}}^{*}\right)$ and $\left({\mathcal{B}}^{*}\right)$ the same computation is carried out with (16) in place of (15). Since $\delta -\left|\gamma \right|=\chi$ and $min\{\nu ,\delta \}-\left|\gamma \right|=\chi$ for $\gamma <0$, the integrand of $\mathcal{J}(t;s)$ becomes

$${\mathsf{\Lambda }}_{\infty }[c_{\ell }/\delta ]u^{-1-\chi }-{\mathsf{\Lambda }}_{\infty }[c_L/\nu ]u^{-1-\delta }+\mathcal{O}\left(u^{-1-\chi -{\rho }_{*}}\right),$$

and integration from $\tau (t;s)$ to infinity gives

$$\mathcal{J}(t;s)={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }{c}_{\ell }}{\delta \chi }}{\displaystyle \frac{1}{{\tau }^{\chi }(t;s)}}(1+\mathcal{O}\left({\tau }^{-{\rho }_1}(t;s)\right))\mathrm{as}t\to \infty ,$$

(65)

because the second term is of relative order ${\tau }^{-(\delta -\chi )}(t;s)={\tau }^{-\left|\gamma \right|}(t;s)$. The passage to ${\left(\nu L_{\infty }t\right)}^{\chi /\nu }$ costs a relative $\mathcal{O}(lnt/t)$ by (29) and a relative $\mathcal{O}(1/t)$, uniform on $[0,d]$, by the consequence (34) of Lemma 5; both are of smaller order than $t^{-{\rho }_1/\nu }$, since ${\rho }_1\le \left|\gamma \right|<\nu$. This gives (62).

For $\mathcal{D}(t;s)$ we use Lemma 5. By (34),

$${\left[\tau (t;s)\right]}^{\left|\gamma \right|}-{\left[\tau \left(t\right)\right]}^{\left|\gamma \right|}=\left|\gamma \right|L_{\infty }M\left(s\right){\left[\tau \left(t\right)\right]}^{\left|\gamma \right|-\nu }(1+o\left(1\right))$$

and therefore, by $\left|\gamma \right|-\nu =-\delta$ and (57),

$$\mathcal{D}(t;s)=-{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }{L}_{\infty }M\left(s\right)}{{\left(\nu L_{\infty }t\right)}^{\delta /\nu }}}(1+o\left(1\right))\mathrm{as}t\to \infty .$$

(66)

Since $\chi <\delta$, the quantity $\mathcal{D}(t;s)$ is of smaller order than $\mathcal{J}(t;s)$ uniformly on $[0,d]$, and (58) gives

$$\zeta (t;s)=1-exp\left\{\mathcal{D}\right(t;s)-\mathcal{J}(t;s\left)\right\},$$

whence (60) and, at leading order, (61). Three quantities are of smaller order than that leading term and together make up the remainder of (62). By (66) the relative size of $\mathcal{D}(t;s)$ is $\mathcal{O}\left(t^{-\left|\gamma \right|/\nu }\right)$, since $\delta -\chi =\left|\gamma \right|$. The relative error of $\mathcal{J}(t;s)$ itself is $\mathcal{O}\left(t^{-min\{{\rho }_{*},\left|\gamma \right|\}/\nu }\right)$ by (65). And the quadratic term of $1-e^{-x}=x-x^2/2+\mathcal{O}\left(x^3\right)$ as $x\to 0$ contributes a relative $\mathcal{O}\left(\mathcal{J}(t;s)\right)=\mathcal{O}\left(t^{-\chi /\nu }\right)$; this is why $\chi$ stands in ${\rho }_1$ and cannot be dropped from it, and it is also the term that the first-order statement (61) absorbs in its $o\left(1\right)$. Of these only the contribution of $\mathcal{D}(t;s)$ depends on s, through $M\left(s\right)$, and it is of the exact relative order $t^{-\left|\gamma \right|/\nu }$; the relative deviation therefore depends on s only through terms of relative order $t^{-\left|\gamma \right|/\nu }$ and smaller. When ${\rho }_1=\left|\gamma \right|$, as for Model T of Section 10, this dependence sits at the order of the remainder in (62) itself, and Figure 1 displays it.

It remains to prove (63). The branching property and the Kolmogorov–Chapman equation give $F(t+u;s)=F(u;F(t;s\left)\right)$ and hence, from (1),

$$\mathcal{P}(t+u;s)=\mathcal{P}(t;s)\mathcal{P}(u;F(t;s\left)\right).$$

Now $\mathsf{\Theta }(t+u)-\mathsf{\Theta }\left(u\right)\to 0$ as $u\to \infty$ for each fixed $t\in \mathcal{T}$. Indeed, the same semigroup relation at $s=0$ gives $\tau (t+u)=\tau (u;F(t;0\left)\right)$, so that by (57) one has $\mathsf{\Theta }(t+u)-\mathsf{\Theta }\left(u\right)=-\mathcal{D}(u;F(t;0\left)\right)$, and this tends to zero by (66) read at the fixed argument $F(t;0)<1$. Multiplying the last identity by $e^{\mathsf{\Theta }\left(u\right)}$ and letting $u\to \infty$ therefore yields

$$\mathcal{P}(t;s)={\displaystyle \frac{\mathcal{U}\left(s\right)}{\mathcal{U}\left(F\right(t;s\left)\right)}}.$$

(67)

That $\mathcal{U}(*)$ is analytic on the whole unit disc, and not merely on $[0,1)$, is a matter of calculus and not of the limit theorem: the substitution of Lemma 7 carries the exponent of (59) into $-{\int }_0^s[h\left(y\right)/f\left(y\right)]\mathbf{d}y$, up to the additive constant $ln{u}_0$ of (70) below, and $1/f(*)$ is analytic on the disc by Lemma 9 below, whose proof uses nothing from the present section, so that the forward reference is not circular. Expanding the denominator of (67) as

$$\mathcal{U}\left(F(t;s)\right)=\sum _{i\in \mathcal{E}}{u}_i{\left[F(t;s)\right]}^i$$

and invoking (1) once more, the double series below being absolutely convergent because ${\sum }_i\left|u_i\right|{\varrho }^i<\infty$ for every $\varrho <1$, while $F(t;s)\le F(t;d)<1$ for $s\in [0,d]$,

$$\sum _{j\in \mathcal{E}}{u}_j{s}^j=\mathcal{U}\left(s\right)=\sum _{i\in \mathcal{E}}{u}_i{\left[F(t;s)\right]}^i\mathcal{P}(t;s)=\sum _{i\in \mathcal{E}}{u}_i{\mathcal{P}}_i(t;s)=\sum _{j\in \mathcal{E}}\left[\sum _{i\in \mathcal{E}}{u}_i{p}_{ij}\left(t\right)\right]s^j,$$

and comparison of the coefficients gives (63). That the $u_j$ are nonnegative is not read off from (59), whose exponent has no sign; it follows from $\mathcal{U}=u_0\mathsf{r}$ of Theorem 9, the coefficients of $\mathsf{r}(*)$ being limits of the nonnegative ratios $p_{0j}\left(t\right)/p_{00}\left(t\right)$. □

Remark 7. 

The hypothesis $c_{\ell }>0$ in Theorem 7 is not cosmetic, and what replaces (61) without it is the following. Let the Basic Axioms hold with $\gamma <0$ and $\chi >0$, let $c_{\ell }=0$, and let $\left({\mathcal{B}}^{*}\right)$ hold with ${\rho }_{\ell }>\left|\gamma \right|$. Then, uniformly in $s\in [0,d]$ for every $d<1$,

$$\zeta (t;s)={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{{\left[\tau \left(t\right)\right]}^{\delta }}}[L_{\infty }M\left(s\right)-{\displaystyle \frac{c_L}{\nu \delta }}]+o\left({\tau }^{-\delta }\left(t\right)\right)\mathrm{as}t\to \infty ,$$

(68)

the remainder being additive because the bracket vanishes at the value of s where the sign turns. Indeed, at $c_{\ell }=0$ hypothesis $\left({\mathcal{B}}^{*}\right)$ gives ${\ell }_{\infty }-\ell \left(x\right)=\mathcal{O}\left(x^{-\delta -{\rho }_{\ell }}\right)$, which is $o\left(x^{-\nu }\right)$ because ${\rho }_{\ell }>\left|\gamma \right|=\nu -\delta$, so that with (8) the relation (15) sharpens to

$${\mathsf{\Lambda }}_{\infty }-\mathsf{\Lambda }\left(x\right)=-{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }{c}_L}{\nu x^{\nu }}}+o\left(x^{-\nu }\right)\mathrm{as}x\to \infty .$$

The term of order $u^{-1-\chi }$ in the integrand of $\mathcal{J}(t;s)$ therefore disappears and only the second one survives, and the computation behind (64) leaves

$$\mathcal{J}(t;s)=-{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }{c}_L}{\nu \delta }}{\displaystyle \frac{1}{{\left[\tau (t;s)\right]}^{\delta }}}+o\left({\tau }^{-\delta }(t;s)\right)\mathrm{as}t\to \infty .$$

(69)

Without the quantitative hypothesis the remainder $o\left(x^{-\delta }\right)$ of (15) would, after division by $x^{1-\left|\gamma \right|}$, be of the larger order $o\left(x^{-1-\chi }\right)$ and would swamp the surviving term. By (66) this is of the same order as $\mathcal{D}(t;s)$ itself rather than of a larger one. The two therefore compete, and subtracting (69) from (66) in $\zeta (t;s)=1-exp\left\{\mathcal{D}\right(t;s)-\mathcal{J}(t;s\left)\right\}$ gives (68), the replacement of $\tau (t;s)$ by $\tau \left(t\right)$ costing $\mathcal{O}({\tau }^{-\delta }\left(t\right)/t)$ uniformly on $[0,d]$ by (34). What is lost at $c_{\ell }=0$ is thus the conclusion of Theorem 7 that the leading term is free of s: the deviation is then governed by $M(*)$, and its sign turns with s, the approach being from above while $M\left(s\right)<c_L/\left(\nu \delta L_{\infty }\right)$ and from below beyond that value. The limit (60) and the identity (58) behind it are untouched.

At $s=0$ the term $\mathcal{D}(t;s)$ vanishes identically, so (58) becomes an exact formula for the return probability.

Theorem 8. 

Under the hypotheses of Lemma 8, for every $t\in \mathcal{T}$

$$e^{\mathsf{\Theta }\left(t\right)}{p}_{00}\left(t\right)=u_{0}exp\{-{\mathcal{J}}_0\left(t\right)\},\mathrm{where}{u}_0=exp\{{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\left|\gamma \right|}}+{\mathcal{K}}_0\}$$

(70)

with ${\mathcal{K}}_0:=\mathcal{K}\left(0\right)$ and ${\mathcal{J}}_0\left(t\right):=\mathcal{J}(t;0)$. If moreover $c_{\ell }>0$, then

$${\mathcal{J}}_0\left(t\right)={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }{c}_{\ell }}{\delta \chi }}{\displaystyle \frac{1}{{\left(\nu L_{\infty }t\right)}^{\chi /\nu }}}(1+o\left(1\right))\mathrm{as}t\to \infty ,$$

(71)

the $o\left(1\right)$ being $\mathcal{O}\left(t^{-{\rho }_1/\nu }\right)$ with ${\rho }_1$ of (62) under $\left({\mathcal{A}}^{*}\right)$ and $\left({\mathcal{B}}^{*}\right)$.

Proof. 

Put $s=0$ in (58); then $\tau (t;0)=\tau \left(t\right)$, so $\mathcal{D}(t;0)=0$ and $\mathcal{U}\left(0\right)=u_0$. Both forms of (71) are (64) and (65) read at $s=0$. □

The invariant measure can also be obtained from the ratio limit property, and the two constructions agree.

Theorem 9. 

Under the hypotheses of Lemma 8 the limits

$${\mathsf{r}}_j:=\underset{t\to \infty }{lim}{\displaystyle \frac{p_{0j}\left(t\right)}{p_{00}\left(t\right)}}$$

exist, are finite and positive for every $j\in \mathcal{E}$, and their generating function is

$$\mathsf{r}\left(s\right)=\sum _{j\in \mathcal{E}}{\mathsf{r}}_j{s}^j=exp\left\{{\int }_1^{1/(1-s)}{\displaystyle \frac{\mathsf{\Lambda }\left(u\right)}{u^{1-\left|\gamma \right|}}}\mathbf{d}u\right\}.$$

(72)

Moreover $\mathcal{U}\left(s\right)=u_0\mathsf{r}\left(s\right)$ identically on $[0,1)$, so that the two invariant measures $\left\{u_j\right\}$ and $\left\{{\mathsf{r}}_j\right\}$ coincide up to the factor $u_0$; and

$$\mathsf{r}\left(s\right)={\displaystyle \frac{1}{u_0}}exp\left\{{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\left|\gamma \right|}}{\displaystyle \frac{1}{{(1-s)}^{\left|\gamma \right|}}}\right\}(1+\mathcal{O}\left({(1-s)}^{\chi }\right))\mathrm{as}s\uparrow 1.$$

(73)

Proof. 

By (51) the ratio $\mathsf{r}(t;s):=\mathcal{P}(t;s)/\mathcal{P}(t;0)$ satisfies

$$ln\mathsf{r}(t;s)={\int }_1^{1/(1-s)}{\displaystyle \frac{\mathsf{\Lambda }\left(u\right)}{u^{1-\left|\gamma \right|}}}\mathbf{d}u-{\int }_{\tau \left(t\right)}^{\tau (t;s)}{\displaystyle \frac{\mathsf{\Lambda }\left(u\right)}{u^{1-\left|\gamma \right|}}}\mathbf{d}u.$$

The second integral tends to zero, because $\tau (t;s)/\tau \left(t\right)\to 1$ as $t\to \infty$ while the integrand is regularly varying of index $\left|\gamma \right|-1<0$ and

$${\tau }^{\left|\gamma \right|}(t;s)-{\tau }^{\left|\gamma \right|}\left(t\right)=\mathcal{O}\left({\tau }^{-\delta }\left(t\right)\right)\mathrm{as}t\to \infty$$

by (66). This proves (72) for the generating functions; since the coefficients $p_{0j}\left(t\right)/p_{00}\left(t\right)$ are nonnegative and the limit (72) is a power series with radius of convergence 1, the continuity theorem for generating functions gives the coefficientwise convergence $p_{0j}\left(t\right)/p_{00}\left(t\right)\to {\mathsf{r}}_j$ as $t\to \infty$.

For the identity $\mathcal{U}=u_0\mathsf{r}$ subtract the exponents. By (59) and (70),

$$ln{\displaystyle \frac{\mathcal{U}\left(s\right)}{u_0}}={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\left|\gamma \right|}}[{\displaystyle \frac{1}{{(1-s)}^{\left|\gamma \right|}}}-1]+\mathcal{K}\left(s\right)-{\mathcal{K}}_0={\int }_1^{1/(1-s)}{\displaystyle \frac{\mathsf{\Lambda }\left(u\right)}{u^{1-\left|\gamma \right|}}}\mathbf{d}u,$$

because

$${\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\left|\gamma \right|}}[{\displaystyle \frac{1}{{(1-s)}^{\left|\gamma \right|}}}-1]={\int }_1^{1/(1-s)}{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{u^{1-\left|\gamma \right|}}}\mathbf{d}u$$

and

$$\mathcal{K}\left(s\right)-{\mathcal{K}}_0=-{\int }_1^{1/(1-s)}{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }-\mathsf{\Lambda }\left(u\right)}{u^{1-\left|\gamma \right|}}}\mathbf{d}u.$$

This is precisely $ln\mathsf{r}\left(s\right)$. Strict positivity of the ${\mathsf{r}}_j$ follows in three steps, none of which uses it. The limits ${\mathsf{r}}_j$ are nonnegative, being limits of the ratios $p_{0j}\left(t\right)/p_{00}\left(t\right)$; hence so are the coefficients $u_j=u_0{\mathsf{r}}_j$, with $u_0>0$ by (70); and the invariance (63), which is proved in Theorem 7 from the identity (67) alone, therefore gives $u_0{\mathsf{r}}_j=u_j\ge u_0{p}_{0j}\left(t\right)$ for any fixed positive $t\in \mathcal{T}$, while $p_{0j}\left(t\right)>0$ for every $j\in \mathcal{E}$ because the chain is irreducible on $\mathcal{E}$.

Finally, (73) follows from $\mathcal{U}=u_0\mathsf{r}$ and (59) on noting that $\mathcal{K}\left(s\right)=\mathcal{O}\left({(1-s)}^{\chi }\right)$ as $s\uparrow 1$, which is the tail estimate (64) read with $1/(1-s)$ in place of $\tau (t;s)$, the two integrals of (56) having one and the same integrand. □

7. The Boundary Regime $\gamma =0$

On the boundary $\delta =\nu$ the exponent in (51) diverges logarithmically, and the normalisation is a power of $\tau \left(t\right)$ whose exponent is ${\mathsf{\Lambda }}_{\infty }$. Put

$$\mathcal{G}\left(X\right):={\int }_X^{\infty }{\displaystyle \frac{\mathsf{\Lambda }\left(u\right)-{\mathsf{\Lambda }}_{\infty }}{u}}\mathbf{d}u.$$

(74)

The integral (74) converges absolutely by (15), the integrand being $\mathcal{O}\left(u^{-1-\nu }\right)$ when $\delta =\nu$.

Theorem 10. 

Let the Basic Axioms hold with $\gamma =0$. Then, uniformly in $s\in [0,d]$ for every $d<1$,

$${\left[\tau \left(t\right)\right]}^{{\mathsf{\Lambda }}_{\infty }}\mathcal{P}(t;s)={\mathcal{U}}_0\left(s\right)\{1-{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\nu t}}[M\left(s\right)+{\displaystyle \frac{c_{\ell }-c_L}{{\nu }^2{L}_{\infty }}}]+o(1/t)\}\mathrm{as}t\to \infty ,$$

(75)

and, if in addition $\left({\mathcal{A}}^{*}\right)$ and $\left({\mathcal{B}}^{*}\right)$ hold, the $o(1/t)$ of (75) is

$$\mathcal{O}\left({\displaystyle \frac{lnt}{t^{1+{\rho }_{*}/\nu }}}\right)\mathrm{as}t\to \infty ,$$

(76)

uniformly in $s\in [0,d]$, with ${\rho }_{*}$ of (6). The limiting generating function is

$${\mathcal{U}}_0\left(s\right):={\displaystyle \frac{1}{{(1-s)}^{{\mathsf{\Lambda }}_{\infty }}}}exp\{-\mathcal{G}\left({\displaystyle \frac{1}{1-s}}\right)\}.$$

(77)

The coefficients of its power series expansion are nonnegative and generate an invariant measure for the transition functions. In particular, with $\lambda :={\mathsf{\Lambda }}_{\infty }/\nu$,

$${\left(\nu L_{\infty }t\right)}^{\lambda }\mathcal{P}(t;s)⟶{\mathcal{U}}_0\left(s\right)\mathrm{as}t\to \infty .$$

(78)

Proof. 

Set $\gamma =0$ in (51) and split the integrand as before:

$$ln\mathcal{P}(t;s)=-{\mathsf{\Lambda }}_{\infty }ln\left[(1-s)\tau (t;s)\right]-\mathcal{G}\left({\displaystyle \frac{1}{1-s}}\right)+\mathcal{G}\left(\tau (t;s)\right),$$

whence, by (77),

$$ln{\displaystyle \frac{{\left[\tau \left(t\right)\right]}^{{\mathsf{\Lambda }}_{\infty }}\mathcal{P}(t;s)}{{\mathcal{U}}_0\left(s\right)}}=-{\mathsf{\Lambda }}_{\infty }ln{\displaystyle \frac{\tau (t;s)}{\tau \left(t\right)}}+\mathcal{G}\left(\tau (t;s)\right).$$

(79)

Both terms are of order $1/t$ and both are computed from the identities already established: the first of them is (35) of Lemma 5. For the second term of (79) insert (15) with $\delta =\nu$, which reads

$$\mathsf{\Lambda }\left(u\right)-{\mathsf{\Lambda }}_{\infty }=-{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }[c_{\ell }-c_L]}{\nu u^{\nu }}}+o\left(u^{-\nu }\right),$$

and integrate:

$$\mathcal{G}\left(\tau (t;s)\right)=-{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }[c_{\ell }-c_L]}{{\nu }^2{\tau }^{\nu }(t;s)}}+o\left({\tau }^{-\nu }(t;s)\right)=-{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }[c_{\ell }-c_L]}{{\nu }^3{L}_{\infty }t}}+o(1/t).$$

Adding the two contributions and exponentiating gives (75).

Under $\left({\mathcal{A}}^{*}\right)$ and $\left({\mathcal{B}}^{*}\right)$ both $o\left(1\right)$ above become explicit, and (76) follows. In the first contribution (29) gives that $1/{\tau }^{\nu }\left(t\right)$ equals $1/\left(\nu L_{\infty }t\right)$ up to a relative $\mathcal{O}(lnt/t)$; in the second, (16) with $\delta =\nu$ replaces the factor $1+o\left(1\right)$ in the integrand of $\mathcal{G}(*)$ by $1+\mathcal{O}\left(u^{-{\rho }_{*}}\right)$, and integration turns this into a relative $\mathcal{O}\left({\tau }^{-{\rho }_{*}}(t;s)\right)$, that is $\mathcal{O}\left(t^{-{\rho }_{*}/\nu }\right)$, uniformly on $[0,d]$. The quadratic term of the exponential contributes $\mathcal{O}(1/t^2)$, which is covered because ${\rho }_{*}\le \nu$. Finally (78) follows from (28). The invariance of the coefficients of ${\mathcal{U}}_0\left(s\right)$ is proved exactly as (63), the only point to check being that ${\mathsf{\Lambda }}_{\infty }ln[\tau (t+u)/\tau \left(u\right)]\to 0$ as $u\to \infty$ for fixed $t\in \mathcal{T}$, which is immediate from (28). Their nonnegativity follows, as the nonnegativity of the $u_j$ followed in the transient regime from Theorem 9, on dividing (78) at s by the same relation at $s=0$: this gives ${\mathcal{U}}_0\left(s\right)={\mathcal{U}}_0\left(0\right)\mathsf{r}\left(s\right)$ with $\mathsf{r}(*)$ of (96), whose coefficients are limits of the nonnegative ratios $p_{0j}\left(t\right)/p_{00}\left(t\right)$ by Theorem 13. That theorem rests on Lemmas 10 and 11 and on Theorems 2 and 5 alone, so it uses nothing from the present section and the forward reference is not circular. □

Remark 8. 

Formula (78) is the exact analogue, on the whole boundary stratum, of the classical statement that $t^{\lambda }\mathcal{P}(t;s)$ converges. At $\nu =\delta =1$, $L\left(x\right)\equiv a_0$, $\ell \left(x\right)\equiv h^{\prime }(1-)$ one has $\lambda =h^{\prime }(1-)/a_0$, $c_L=c_{\ell }=0$, $\mathcal{G}\left(X\right)\equiv 0$ and ${\mathcal{U}}_0\left(s\right)={(1-s)}^{-\lambda }$, and (78) reduces to ${\left(a_{0}t\right)}^{\lambda }\mathcal{P}(t;s)\to {(1-s)}^{-\lambda }$ as $t\to \infty$, in agreement with [30,35]. The novelty for $\nu <1$ is that the normalisation is $t^{{\mathsf{\Lambda }}_{\infty }/\nu }$ rather than $t^{{\mathsf{\Lambda }}_{\infty }}$, and that the limit acquires the correction factor $exp\{-\mathcal{G}\}$ generated by the joint remainder of the two slowly varying functions. Note also that (75) is a rate of convergence on the boundary, and that the rate is $1/t$ irrespective of the position of ${\mathsf{\Lambda }}_{\infty }$ relative to the threshold ν: the classification of Theorem 6 concerns the size of the limit, not the speed at which it is approached.

8. The Positive Recurrent Regime $\gamma >0$

For $\gamma >0$ the exponent in (51) converges and no normalisation is needed at all: the transition functions converge to a proper stationary distribution. Put

$$\mathcal{W}\left(X\right):={\int }_X^{\infty }{\displaystyle \frac{\mathsf{\Lambda }\left(u\right)}{u^{1+\gamma }}}\mathbf{d}u,$$

(80)

which is finite for every $X\ge 1$ and satisfies

$$\mathcal{W}\left(X\right)={\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\gamma }}{\displaystyle \frac{1}{X^{\gamma }}}(1+o\left(1\right))\mathrm{as}X\to \infty ,$$

(81)

because $\mathsf{\Lambda }\left(u\right)\to {\mathsf{\Lambda }}_{\infty }$ as $u\to \infty$ by (13).

Neither the limit in this regime nor the constant in front of the rate is new; the two sources are named in Section 1. The limit is the stationary generating function $\pi \left(s\right)$ of Theorem 11 below, written in those sources in the equivalent form $exp\left\{{\int }_s^1[h\left(u\right)/f\left(u\right)]\mathbf{d}u\right\}$, and the relative deviation is given there as

$${\displaystyle \frac{\ell \left(\tau \right(t\left)\right)}{\gamma L^{\delta /\nu }\left(\tau \left(t\right)\right){[\nu t+1/\mathcal{V}(1-s)]}^{\gamma /\nu }}},$$

which under the present hypotheses is exactly the leading term of the rate obtained in that theorem: the two slowly varying factors converge, and ${\ell }_{\infty }/\left(\gamma L_{\infty }^{\delta /\nu }\right)={\mathsf{\Lambda }}_{\infty }/\left(\gamma L_{\infty }^{\gamma /\nu }\right)$ because $\delta /\nu =1+\gamma /\nu$. What the theorem below adds is that those factors are replaced by the constants they converge to, so that the rate is a power of t with a numerical coefficient; that the approach is monotone; that it is uniform in s on the whole of $[0,1)$ when the rate is expressed through $\tau (t;s)$; and, through Section 9, the same rate for the individual transition probabilities. In those two sources the passage from the generating function to the coefficients is made by the continuity theorem for power series, which yields the convergence of the coefficients but not a rate for them; supplying the rate is the object of Section 9.

Theorem 11. 

Let the Basic Axioms hold with $\gamma >0$. Then the stationary distribution $\{{\pi }_j,j\in \mathcal{E}\}$ of the MBIS exists, its generating function is

$$\pi \left(s\right)=\sum _{j\in \mathcal{E}}{\pi }_j{s}^j=exp\{-\mathcal{W}\left({\displaystyle \frac{1}{1-s}}\right)\},$$

(82)

and the approach to it is monotone with the exact rate

$$\mathcal{P}(t;s)=\pi \left(s\right)\{1+{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\gamma }}{\displaystyle \frac{1}{{\left[\tau (t;s)\right]}^{\gamma }}}(1+o\left(1\right))\}\mathrm{as}t\to \infty ,$$

(83)

uniformly in $s\in [0,1)$. In terms of t alone,

$$\mathcal{P}(t;s)=\pi \left(s\right)\{1+{\displaystyle \frac{{\mathsf{\Lambda }}_{\infty }}{\gamma }}{\displaystyle \frac{1}{{\left(\nu L_{\infty }t\right)}^{\gamma /\nu }}}(1+o\left(1\right))\}\mathrm{as}t\to \infty ,$$

(84)

uniformly in $s\in [0,d]$ for every $d<1$; on the whole of $[0,1)$ the factor $1+o\left(1\right)$ in (84) must be replaced by a bounded one.

Proof. 

By (51) and (80),

$$ln\mathcal{P}(t;s)=-\mathcal{W}\left({\displaystyle \frac{1}{1-s}}\right)+\mathcal{W}\left(\tau (t;s)\right),$$

(85)

an exact identity. Since (85) makes $\mathcal{P}(t;s)$ a function of $\tau (t;s)$ alone, and since $\tau (t;s)\uparrow \infty$ as $t\to \infty$ and $\mathcal{W}(*)$ is positive and decreasing, $\mathcal{P}(t;s)$ decreases to $\pi \left(s\right)$ given by (82). The function $\pi (*)$ is a power series with nonnegative coefficients, and $\pi (1-)=1$ because $\mathcal{W}\left(X\right)\to 0$ as $X\to \infty$; so it is the generating function of a proper probability distribution on $\mathcal{E}$, which by Theorem 6(i) is the stationary distribution. For the rate use (81) at $X=\tau (t;s)$ together with $e^x=1+x+\mathcal{O}\left(x^2\right)$ as $x\to 0$; since $\tau (t;s)\ge \tau \left(t\right)\to \infty$ as $t\to \infty$ uniformly in $s\in [0,1)$ by Theorem 2, both relations hold uniformly there, and (83) follows.

Replacing $\tau (t;s)$ by ${\left(\nu L_{\infty }t\right)}^{1/\nu }$ requires $\tau (t;s)/\tau \left(t\right)\to 1$ as $t\to \infty$, which by (34) holds uniformly on $[0,d]$ for every $d<1$ but not on $[0,1)$: along any $s=s\left(t\right)\uparrow 1$ with $M\left(s\right(t\left)\right)/t\to 1$ one has $\tau (t;s)/\tau \left(t\right)\to 2^{1/\nu }$, and the constant of (84) acquires the factor $1/2^{\gamma /\nu }$. This proves (84) on $[0,d]$; the bounded form of (84) holds on the whole of $[0,1)$ because $\mathcal{W}\left(\tau \right(t;s\left)\right)\le \mathcal{W}\left(\tau \right(t\left)\right)$ there. □

Remark 9. 

The three regimes fit together as they should. Comparing (59), (77) and (82) one sees that in every case the limiting generating function comes from the same exponent $-{\int }_{1/(1-s)}^{\infty }\mathsf{\Lambda }\left(u\right)u^{-1-\gamma }\mathbf{d}u$ of (51), the constant part ${\mathsf{\Lambda }}_{\infty }$ of $\mathsf{\Lambda }(*)$ being split off and integrated separately whenever $\gamma \le 0$, which is exactly when it makes the integral diverge. The rate of convergence is in each case governed by the tail of the same integral: it is ${\tau }^{-\chi }\left(t\right)$ for $\gamma <0$, ${\tau }^{-\nu }\left(t\right)$ for $\gamma =0$ and ${\tau }^{-\gamma }\left(t\right)$ for $\gamma >0$. Only the transient regime requires a condition beyond the Basic Axioms, namely $\chi >0$, and only there does the normalising factor grow.

9. From the Generating Functions to the Transition Probabilities

Every rate established so far is a rate for a generating function on a real interval. Passing from it to a rate for the individual transition probabilities is not automatic. The coefficients of the difference of two power series with nonnegative coefficients have no sign, so a bound on that difference along $[0,d]$ carries no information about them one at a time. The customary remedy is to assume that the ratios converge monotonically in t, which makes those coefficients nonnegative and the transfer elementary. We do not assume it. The systematic device for passing from the behaviour of a generating function near its singularity to the asymptotics of its coefficients is the singularity analysis of Flajolet and Odlyzko [9], but it requires an analytic continuation beyond the disc of convergence, which is not available here. What is needed instead is the same bound for complex s on the circle $\left|s\right|=d$, and criticality supplies it through the factorisation recorded next.

The passage from the real interval to the disc costs no new hypothesis. The functions that are carried across it are power series with nonnegative coefficients, analytic on $\left|s\right|<1$ as recorded in Section 2; an identity between two of them proved for real $s\in [0,1)$ therefore holds throughout $\left|s\right|<1$ by the identity theorem, and we say so at each of the two places below where that extension is used. Objects defined by an integral along the real axis, such as $\mathcal{K}\left(s\right)$ of (56) or $\mathcal{W}(*)$ of (80), are not carried across and continue to occur with a real argument only. The regular variation hypotheses $\left[f_{\nu }\right]$ and $\left[h_{\delta }\right]$, for their part, are, and remain, statements about a real argument: they are never evaluated off the real axis, and they enter the complex estimates below only through (88), which is read on the real segment $z=1-y$ with $y\in (0,1]$. Off that segment the power series structure alone is used.

Lemma 9. 

Let the system be critical. Then, with ${\sigma }_i\left(z\right):=1+z+\dots +z^{i-1}$,

$$f\left(z\right)={(1-z)}^2\mathsf{\Omega }\left(z\right)\mathrm{and}h\left(z\right)=-(1-z)H\left(z\right),$$

(86)

where

$$\mathsf{\Omega }\left(z\right):=\sum _{k\ge 2}{a}_k{\displaystyle \sum _{i=1}^{k-1}}{\sigma }_i\left(z\right)\mathrm{and}H\left(z\right):=\sum _{k\in \mathbb{N}}{b}_k{\sigma }_k\left(z\right)$$

(87)

are power series with nonnegative coefficients, convergent for $\left|z\right|<1$. Moreover $f\left(z\right)\ne 0$ and $\mathsf{\Omega }\left(z\right)\ne 0$ throughout $\left|z\right|<1$, and for real $y\in (0,1]$

$$\mathsf{\Omega }(1-y)={\displaystyle \frac{L(1/y)}{y^{1-\nu }}}\mathrm{and}H(1-y)={\displaystyle \frac{\ell (1/y)}{y^{1-\delta }}}.$$

(88)

Consequently $M\left(s\right)={\int }_0^s[1/f\left(x\right)]\mathbf{d}x$ is single-valued and analytic on $\left|s\right|<1$, and for each $d<1$

$$c_d:=\underset{\left|z\right|\le d}{min}\left|f\left(z\right)\right|>0.$$

(89)

Proof. 

Criticality means $f\left(1\right)={\sum }_k{a}_k=0$ and $f^{\prime }(1-)={\sum }_{k}k{a}_k=0$. From the first,

$$f\left(z\right)=\sum _k{a}_k(z^k-1)=-(1-z)\sum _{k\in \mathbb{N}}{a}_k{\sigma }_k\left(z\right),$$

so that

$${\displaystyle \frac{f\left(z\right)}{1-z}}=-a_1-\sum _{k\ge 2}{a}_k{\sigma }_k\left(z\right).$$

From the second, $-a_1={\sum }_{k\ge 2}k{a}_k={\sum }_{k\ge 2}{a}_k{\sigma }_k\left(1\right)$, whence

$${\displaystyle \frac{f\left(z\right)}{1-z}}=\sum _{k\ge 2}{a}_k[{\sigma }_k\left(1\right)-{\sigma }_k\left(z\right)].$$

Since

$${\sigma }_k\left(1\right)-{\sigma }_k\left(z\right)={\displaystyle \sum _{i=1}^{k-1}}(1-z^i)=(1-z){\displaystyle \sum _{i=1}^{k-1}}{\sigma }_i\left(z\right),$$

the first half of (86) follows with $\mathsf{\Omega }(*)$ as in (87); all its coefficients are nonnegative because $a_k\ge 0$ for $k\ge 2$. The statement for $h(*)$ is the same computation with ${\sum }_k{b}_k=0$, and there the first step already suffices.

For the zeros, let $\left|z\right|<1$. Then

$$\left|{\sigma }_k\left(z\right)\right|\le {\sigma }_k\left(\left|z\right|\right)<{\sigma }_k\left(1\right)$$

for every $k\ge 2$, so that

$$\left|\sum _{k\ge 2}{a}_k{\sigma }_k\left(z\right)\right|\le \sum _{k\ge 2}{a}_k{\sigma }_k\left(\left|z\right|\right)<\sum _{k\ge 2}{a}_k{\sigma }_k\left(1\right)=-a_1,$$

the strict inequality holding because $a_k>0$ for at least one $k\ge 2$, which the consistency condition $a_0<-a_1$ forces since ${\sum }_{k\ge 2}{a}_k=-a_1-a_0>0$. Hence $f\left(z\right)/(1-z)=-a_1-{\sum }_{k\ge 2}{a}_k{\sigma }_k\left(z\right)$ is nonzero, and so are $f\left(z\right)$ and $\mathsf{\Omega }\left(z\right)=f\left(z\right)/{(1-z)}^2$. Identities (88) are $\left[f_{\nu }\right]$ and $\left[h_{\delta }\right]$ divided by $y^2$ and by y respectively. The disc is simply connected and $1/f\left(z\right)$ is analytic on it, so $M\left(s\right)$ is single-valued and analytic; and (89) holds because $\left|f\right|$ is continuous and positive on the compact disc $\left|z\right|\le d$. □

What is needed besides is that, on $\left|s\right|\le d$, the function $F(t;s)$ stays within a distance of order $q\left(t\right)/t$ of its value at $s=0$. This is what makes all the complex estimates below cheap. Here and in the rest of this section $C_d$ denotes a finite positive constant depending on d alone, not necessarily the same at each occurrence.

Lemma 10. 

Let $\left[f_{\nu }\right]$, $\left[h_{\delta }\right]$ and $\left(\mathcal{A}\right)$ hold, and fix $d<1$. Then, as $t\to \infty$, uniformly in $\left|s\right|\le d$,

$$|F(t;s)-F(t;0)|=\mathcal{O}\left({\displaystyle \frac{q\left(t\right)}{t}}\right)\mathrm{and}{\displaystyle \frac{1-F(t;s)}{q\left(t\right)}}=1+\mathcal{O}(1/t)$$

(90)

and

$${\displaystyle \frac{f\left(F\right(t;s\left)\right)}{f\left(F\right(t;0\left)\right)}}=1+\mathcal{O}(1/t).$$

(91)

Moreover, for all sufficiently large t and every u on the segment joining $F(t;0)$ to $F(t;s)$,

$$\left|{\displaystyle \frac{h\left(u\right)}{f\left(u\right)}}\right|\le C\mathsf{\Lambda }\left(\tau \left(t\right)\right)q^{\gamma -1}\left(t\right)$$

(92)

with a constant C depending on ν and δ only; the threshold beyond which (92) holds depends on d, the constant C does not. Hypothesis $\left[h_{\delta }\right]$ is used for (92) alone; (90) and (91) need the other two only.

Proof. 

Since $F(t;s)-F(t;0)={\sum }_{j\in \mathcal{S}}{\mathsf{p}}_j\left(t\right)s^j$ has nonnegative coefficients,

$$|F(t;s)-F(t;0)|\le \sum _{j\in \mathcal{S}}{\mathsf{p}}_j\left(t\right)d^j={\mathsf{p}}_1\left(t\right)V(t;d),$$

and $V(t;d)\to a_{0}M\left(d\right)<\infty$ by Theorem 5. By (38) and (42), ${\mathsf{p}}_1\left(t\right)/q\left(t\right)=1/\left(a_{0}A\left(t\right)\right)$ with $A\left(t\right)=\nu t(1+o(1\left)\right)$ as $t\to \infty$, which gives the first half of (90); the second half follows from $1-F(t;s)=q\left(t\right)-\left[F\right(t;s)-F(t;0\left)\right]$.

Write $z:=F(t;s)$ and $z_0:=F(t;0)=1-q\left(t\right)$, so that $|z-z_0|\le C_{d}q\left(t\right)/t$ by (90). By (86),

$${\displaystyle \frac{f\left(z\right)}{f\left(z_0\right)}}={\left({\displaystyle \frac{1-z}{1-z_0}}\right)}^2{\displaystyle \frac{\mathsf{\Omega }\left(z\right)}{\mathsf{\Omega }\left(z_0\right)}},$$

and the first factor is $1+\mathcal{O}(1/t)$ by (90). For the second put $\varrho :=q\left(t\right)/2$ and note that $\left|\varpi \right|\le 1-\varrho$ whenever $|\varpi -z_0|\le \varrho$, so that $\left|\mathsf{\Omega }\right(\varpi \left)\right|\le \mathsf{\Omega }(1-\varrho )$ there, the coefficients of $\mathsf{\Omega }(*)$ being nonnegative. Cauchy’s estimate on that disc gives $\left|{\mathsf{\Omega }}^{\prime }\left(\varpi \right)\right|\le 2\mathsf{\Omega }(1-\varrho )/\varrho$ for $|\varpi -z_0|\le \varrho /2$, whence

$$|{\displaystyle \frac{\mathsf{\Omega }\left(z\right)}{\mathsf{\Omega }\left(z_0\right)}}-1|\le {\displaystyle \frac{4|z-z_0|}{q\left(t\right)}}{\displaystyle \frac{\mathsf{\Omega }(1-q(t)/2)}{\mathsf{\Omega }(1-q(t\left)\right)}}.$$

By (88) the last ratio equals

$$2^{1-\nu }{\displaystyle \frac{L\left(2\tau \right(t\left)\right)}{L\left(\tau \right(t\left)\right)}}⟶2^{1-\nu }\mathrm{as}t\to \infty ,$$

and $|z-z_0|/q\left(t\right)$ is $\mathcal{O}(1/t)$. This proves (91).

Finally, on the segment in question u satisfies $|u-z_0|\le C_{d}q\left(t\right)/t$, so that for all large t it lies in the disc $|\varpi -z_0|\le \varrho /2$ and the bound just proved for $\mathsf{\Omega }\left(z\right)/\mathsf{\Omega }\left(z_0\right)$ applies verbatim with u in place of z; hence

$$\left|\mathsf{\Omega }\left(u\right)\right|\ge \frac{1}{2}\mathsf{\Omega }(1-q\left(t\right)).$$

A lower bound of this kind is what is needed here, and it does not follow from the nonnegativity of the coefficients of $\mathsf{\Omega }(*)$, which gives upper bounds only. Together with $|1-u|\ge q\left(t\right)/2$ and, the coefficients of $H(*)$ being nonnegative, $\left|H\right(u\left)\right|\le H(1-q(t)/2)$, we obtain from (86) and (88)

$$\left|{\displaystyle \frac{h\left(u\right)}{f\left(u\right)}}\right|={\displaystyle \frac{\left|H\right(u\left)\right|}{|1-u|\left|\mathsf{\Omega }\right(u\left)\right|}}\le {\displaystyle \frac{4H(1-q(t)/2)}{q\left(t\right)\mathsf{\Omega }(1-q(t\left)\right)}}\le C{\displaystyle \frac{\ell \left(\tau \left(t\right)\right)q^{\delta -1}\left(t\right)}{q\left(t\right)L\left(\tau \left(t\right)\right)q^{\nu -1}\left(t\right)}},$$

the last step using $\ell \left(2\tau \right(t\left)\right)/\ell \left(\tau \right(t\left)\right)\to 1$ as $t\to \infty$, which is (92) because $\gamma =\delta -\nu$. □

We can now transfer the rate of Theorem 5 to the individual local probabilities of the branching system without immigration, which is what legitimises the passage to the coefficients in (50).

Theorem 12. 

Let $\left[f_{\nu }\right]$ and $\left(\mathcal{A}\right)$ hold and let $d<1$. Then there is a constant $C_d<\infty$ such that

$$|{\displaystyle \frac{{\mathsf{p}}_j\left(t\right)}{{\mathsf{p}}_1\left(t\right)}}-a_0{m}_j|\le {\displaystyle \frac{C_d}{d^{j}t}}\mathrm{for}\mathrm{all}j\in \mathcal{S}$$

(93)

and all sufficiently large t.

Proof. 

The relation $\partial F(t;s)/\partial s=f\left(F\right(t;s\left)\right)/f\left(s\right)$ obtained in Lemma 6 holds for real $s\in [0,1)$; both sides are analytic in $\left|s\right|<1$, by Lemma 9 for the denominator, so the identity theorem extends it to the whole disc. The same applies to $M^{\prime }\left(s\right)=1/f\left(s\right)$. Integrating both along the segment from 0 to s and using $a_0{\mathsf{p}}_1\left(t\right)=f\left(F(t;0)\right)$ from (38), we obtain for every $\left|s\right|<1$ the exact identity

$$V(t;s)-a_{0}M\left(s\right)=a_0{\int }_0^s{\displaystyle \frac{1}{f\left(x\right)}}[{\displaystyle \frac{f\left(F\right(t;x\left)\right)}{f\left(F\right(t;0\left)\right)}}-1]\mathbf{d}x.$$

(94)

By (91) the bracket in (94) is $\mathcal{O}(1/t)$ uniformly for $\left|x\right|\le d$, and $\left|f\left(x\right)\right|\ge c_d>0$ there by (89); since the segment has length at most d,

$$\underset{\left|s\right|\le d}{sup}|V(t;s)-a_{0}M\left(s\right)|\le {\displaystyle \frac{C_d}{t}}\mathrm{as}t\to \infty .$$

(95)

Both $V(t;*)$ and $a_{0}M(*)$ are analytic on the disc, so Cauchy’s coefficient formula on the circle $\left|s\right|=d$ turns (95) into (93). □

For the system with immigration the corresponding statement is uniform across the three regimes, because the ratio $\mathsf{r}(t;s)=\mathcal{P}(t;s)/p_{00}\left(t\right)$ of the proof of Theorem 9 obeys an exact identity in which the sign of $\gamma$ plays no part. The generating function $\mathsf{r}\left(s\right)$ of (72) has, by the substitution of Lemma 7, the form

$$\mathsf{r}\left(s\right)=exp\{-{\int }_0^s{\displaystyle \frac{h\left(y\right)}{f\left(y\right)}}\mathbf{d}y\}\mathrm{for}s\in [0,1),$$

(96)

in which $\gamma$ no longer appears; we use (96) for $\mathsf{r}\left(s\right)$ in all three regimes, and for $\gamma >0$ it equals $\pi \left(s\right)/\pi \left(0\right)$ with $\pi (*)$ as in (82).

Lemma 11. 

Let $\left[f_{\nu }\right]$ and $\left[h_{\delta }\right]$ hold. Then for every $\left|s\right|<1$ and $t\in \mathcal{T}$

$${\displaystyle \frac{\mathsf{r}(t;s)}{\mathsf{r}\left(s\right)}}=exp\left\{{\int }_{F(t;0)}^{F(t;s)}{\displaystyle \frac{h\left(u\right)}{f\left(u\right)}}\mathbf{d}u\right\}.$$

(97)

Proof. 

The substitution used in Lemma 7 gives

$$ln\mathcal{P}(t;s)={\int }_s^{F(t;s)}[h\left(u\right)/f\left(u\right)]\mathbf{d}u,$$

first for real s and then, both sides being analytic, for $\left|s\right|<1$. Subtracting the same identity at $s=0$ and adding ${\int }_0^s[h\left(u\right)/f\left(u\right)]\mathbf{d}u$ to both sides gives (97), all integrals being taken along segments inside the disc, where $1/f\left(u\right)$ is analytic. □

Theorem 13. 

Let the Basic Axioms hold and let $d<1$. Then, uniformly in $\left|s\right|\le d$,

$${\displaystyle \frac{\mathsf{r}(t;s)}{\mathsf{r}\left(s\right)}}=1+\mathcal{O}\left({\displaystyle \frac{q^{\gamma }\left(t\right)}{t}}\right)=1+\mathcal{O}\left({\displaystyle \frac{1}{t^{1+\gamma /\nu }}}\right)\mathrm{as}t\to \infty ,$$

(98)

and consequently, with $\mathsf{r}\left(s\right)={\sum }_{j\in \mathcal{E}}{\mathsf{r}}_j{s}^j$, there is a constant $C_d<\infty$ such that

$$|{\displaystyle \frac{p_{0j}\left(t\right)}{p_{00}\left(t\right)}}-{\mathsf{r}}_j|\le {\displaystyle \frac{C_d}{d^j{t}^{1+\gamma /\nu }}}\mathrm{for}\mathrm{all}j\in \mathcal{E}$$

(99)

and all large t. In particular the strong ratio limit property holds in all three regimes of Theorem 6, with one and the same limiting generating function (96).

Proof. 

Combining (97) with (90) and (92), the exponent is bounded in modulus by

$$C_d{\displaystyle \frac{q\left(t\right)}{t}}\mathsf{\Lambda }\left(\tau \left(t\right)\right)q^{\gamma -1}\left(t\right)=\mathcal{O}\left({\displaystyle \frac{q^{\gamma }\left(t\right)}{t}}\right)\mathrm{as}t\to \infty ,$$

because $\mathsf{\Lambda }\left(\tau \left(t\right)\right)\to {\mathsf{\Lambda }}_{\infty }$ as $t\to \infty$. Theorem 2 turns $q^{\gamma }\left(t\right)$ into ${\left(\nu L_{\infty }t\right)}^{-\gamma /\nu }$ up to a factor $1+o\left(1\right)$, which is (98). Since $\mathsf{r}(*)$ is bounded on $\left|s\right|\le d$ by $\mathsf{r}\left(d\right)$, the same bound holds for $\left|\mathsf{r}\right(t;s)-\mathsf{r}(s\left)\right|$ up to a constant, and Cauchy’s coefficient formula on $\left|s\right|=d$ gives (99). □

The three limit theorems of Section 6, Section 7 and Section 8 now become statements about the transition probabilities themselves. In each case the diagonal entry is covered by the corresponding theorem at $s=0$, and (99) carries the result to the whole row.

Corollary 2. 

Let the Basic Axioms hold and let $d<1$. There is a constant $C_d<\infty$ such that for all $j\in \mathcal{E}$ and all large t:

(i)

if $\gamma <0$, $\chi >0$ and $c_{\ell }>0$, then

$$|e^{\mathsf{\Theta }\left(t\right)}{p}_{0j}\left(t\right)-u_j|\le C_d{d}^{-j}{\left(\nu L_{\infty }t\right)}^{-\chi /\nu },$$

(100)

where $\mathcal{U}\left(s\right)={\sum }_{j\in \mathcal{E}}{u}_j{s}^j$ is the generating function (59);

(ii)

if $\gamma =0$, then

$$|{\left[\tau \left(t\right)\right]}^{{\mathsf{\Lambda }}_{\infty }}{p}_{0j}\left(t\right)-u_j^0|\le C_d{d}^{-j}/t,$$

(101)

where ${\mathcal{U}}_0\left(s\right)={\sum }_{j\in \mathcal{E}}{u}_j^0{s}^j$ is given by (77);

(iii)

if $\gamma >0$, then

$$|p_{0j}\left(t\right)-{\pi }_j|\le C_d{d}^{-j}{\left(\nu L_{\infty }t\right)}^{-\gamma /\nu },$$

(102)

where $\pi \left(s\right)={\sum }_{j\in \mathcal{E}}{\pi }_j{s}^j$ is the stationary generating function (82).

Proof. 

Write $\mathsf{N}\left(t\right)$ for the normalising factor of the regime under consideration and ${\xi }_j$ for the limit asserted for the row entries, so that

$$\mathsf{N}\left(t\right)=e^{\mathsf{\Theta }\left(t\right)},{\left[\tau \left(t\right)\right]}^{{\mathsf{\Lambda }}_{\infty }},1\mathrm{and}{\xi }_j=u_j,u_j^0,{\pi }_j$$

in the cases (i), (ii) and (iii) respectively, so that in all three the assertion is a bound on $|\mathsf{N}\left(t\right)p_{0j}\left(t\right)-{\xi }_j|$ at the rate displayed. Multiplying and dividing by the diagonal entry,

$$\mathsf{N}\left(t\right)p_{0j}\left(t\right)=\mathsf{N}\left(t\right)p_{00}\left(t\right)·[p_{0j}\left(t\right)/p_{00}\left(t\right)].$$

In each case ${\xi }_j={\xi }_0{\mathsf{r}}_j$, because dividing the defining limit at s by the same limit at $s=0$ leaves $\mathsf{r}\left(s\right)$ of (96), the normalising factor cancelling. Then

$$|\mathsf{N}\left(t\right)p_{0j}\left(t\right)-{\xi }_j|\le {\mathsf{r}}_j|\mathsf{N}\left(t\right)p_{00}\left(t\right)-{\xi }_0|+\mathsf{N}\left(t\right)p_{00}\left(t\right)|{\displaystyle \frac{p_{0j}\left(t\right)}{p_{00}\left(t\right)}}-{\mathsf{r}}_j|.$$

In the first term ${\mathsf{r}}_j\le \mathsf{r}\left(d\right)d^{-j}$, and $|\mathsf{N}\left(t\right)p_{00}\left(t\right)-{\xi }_0|$ is estimated by Theorem 8 in case (i), by Theorem 10 at $s=0$ in case (ii) and by Theorem 11 at $s=0$ in case (iii), the three rates being $t^{-\chi /\nu }$, $t^{-1}$ and $t^{-\gamma /\nu }$. In the second term $\mathsf{N}\left(t\right)p_{00}\left(t\right)$ is bounded and the factor after it is estimated by (99), giving $d^{-j}{t}^{-1-\gamma /\nu }$. In each of the three cases the first term dominates: in (i) because $\chi <\delta$ and $1+\gamma /\nu =\delta /\nu$, in (ii) because the two rates coincide, and in (iii) because $\gamma /\nu <1+\gamma /\nu$. □

Remark 10. 

Two features of (93) and (99) are worth recording. The factor $d^{-j}$ can be made as close to 1 as desired by taking d close to 1, at the cost of enlarging $C_d$; the bounds are therefore genuinely uniform on finite ranges of j and degrade only geometrically. And the ratio $p_{0j}\left(t\right)/p_{00}\left(t\right)$ settles down faster than the normalised transition functions themselves do, as one expects, since the slowly decaying part of $\mathcal{P}(t;s)$ is common to numerator and denominator and cancels; the gain, however, is not the same in the three regimes. It is the full factor $1/t$ only when $\gamma >0$, where (99) gives $t^{-1-\gamma /\nu }$ against the rate $t^{-\gamma /\nu }$ of (102). When $\gamma <0$ the normalised rate of (100) is $t^{-\chi /\nu }$ and (99) gives $t^{-\delta /\nu }$, a gain of $t^{-\left|\gamma \right|/\nu }$ only; and when $\gamma =0$ the two rates, that of (101) and that of (99), are both $1/t$ and there is no gain at all, which is why the proof of Corollary 2 has to say in case (ii) that the two rates coincide.

10. Numerical Illustrations

Every constant occurring above is computable, and this section verifies them on a family of models in which the two generating functions are given in closed form and the hypotheses can be checked by inspection. Fix $\nu ,\delta \in (0,1]$, $c,\beta >0$ and $\alpha ,\eta \in (0,1)$, and put

$$f\left(s\right)=c[{(1-s)}^{1+\nu }-\alpha {(1-s)}^{1+2\nu }]\mathrm{and}h\left(s\right)=-\beta [{(1-s)}^{\delta }-\eta {(1-s)}^{2\delta }].$$

(103)

The pair (103) is of the form $\left[f_{\nu }\right]$ and $\left[h_{\delta }\right]$ with

$$L\left(x\right)=c[1-{\displaystyle \frac{\alpha }{x^{\nu }}}]\mathrm{and}\ell \left(x\right)=\beta [1-{\displaystyle \frac{\eta }{x^{\delta }}}],$$

(104)

both of them normalised and slowly varying; the index functions of (104) are ${\epsilon }_L\left(x\right)=c\alpha \nu x^{-\nu }/L\left(x\right)$ and ${\epsilon }_{\ell }\left(x\right)=\beta \eta \delta x^{-\delta }/\ell \left(x\right)$. Hence $\left({\mathcal{A}}^{*}\right)$ and $\left({\mathcal{B}}^{*}\right)$ hold with ${\rho }_L=\nu$ and ${\rho }_{\ell }=\delta$, so that ${\rho }_{*}=min\{\nu ,\delta \}$ in (6) and the second-order rates (62) and (76) are available for the family; the constants of the paper are

$$\begin{array}{c}{L}_{\infty }=c,{\ell }_{\infty }=\beta ,c_L=\nu \alpha ,c_{\ell }=\delta \eta ,\\ beginequation5pt]{\mathsf{\Lambda }}_{\infty }={\displaystyle \frac{\beta }{c}},\vartheta ={\displaystyle \frac{\nu \alpha }{c}},a_0=c(1-\alpha ).\end{array}$$

(105)

The list (105) is all that the theorems require. That (103) really are infinitesimal generating functions, that is, that $a_k\ge 0$ for $k\ge 2$, $a_1<0<a_0$ and $b_k\ge 0$ for $k\ge 1$, $b_0<0$, is verified directly from the binomial coefficients for each parameter set of Table 1.

The whole computation reduces to one ordinary differential equation. By the backward equation (2) the function ${\tau }^{\nu }(t;s)=R^{-\nu }(t;s)$ satisfies

$${\displaystyle \frac{\mathbf{d}}{\mathbf{d}t}}{\tau }^{\nu }(t;s)=\nu L\left(\tau (t;s)\right)\mathrm{with}{\tau }^{\nu }(0;s)={\displaystyle \frac{1}{{(1-s)}^{\nu }}},$$

(106)

which involves $L(*)$ alone, the immigration entering only through the quadrature that follows. For the family (104) the right-hand side of (106) is $c\nu [{\tau }^{\nu }(t;s)-\alpha ]/{\tau }^{\nu }(t;s)$, so that separation of the variables solves (106) in closed form and gives the implicit relation

$${\tau }^{\nu }(t;s)+\alpha ln[{\tau }^{\nu }(t;s)-\alpha ]=c\nu t+{\displaystyle \frac{1}{{(1-s)}^{\nu }}}+\alpha ln[{\displaystyle \frac{1}{{(1-s)}^{\nu }}}-\alpha ],$$

(107)

an equation of Lambert type which is inverted to any prescribed accuracy, and in the same way

$$M\left(s\right)={\displaystyle \frac{1}{c\nu }}[{\displaystyle \frac{1}{{(1-s)}^{\nu }}}-1+\alpha ln{\displaystyle \frac{{(1-s)}^{-\nu }-\alpha }{1-\alpha }}].$$

(108)

No numerical integration of (106) is therefore needed anywhere below; all the tables and figures of this section were produced from (107), (108) and the exact formulae (51) and (97), on the four parameter sets of Table 1. The letter in the name of each model marks the regime of Theorem 6 that it illustrates: T stands for transient, B for the boundary $\gamma =0$, on which both the null recurrent and the transient outcome are shown, and R for positive recurrent. Model T is the one for which ${\mathsf{\Lambda }}_{\infty }\ne \left|\gamma \right|$, so that the earlier normalisation $exp\left\{{\tau }^{\left|\gamma \right|}\left(t\right)\right\}$ does not apply to it at all, whereas (55) does.

10.1. The Survival Probability

The agreement in Table 2 is to nine significant figures by $t={10}^5$ and to every digit printed by $t={10}^7$, which leaves no room for doubt about the constant $\mathfrak{c}$ of (30). In the linearised form of (31), the deviation of $q\left(t\right)$ from its leading term ${\left(\nu L_{\infty }t\right)}^{-1/\nu }$, multiplied by ${\nu }^3{L}_{\infty }t$, follows the straight line $c_{L}ln\left(\nu L_{\infty }t\right)-\nu \mathfrak{c}$ in $lnt$ from $t\approx 50$ onward, and changes sign in the process, which no one-term approximation can reproduce.

For this family both constants of (29) are open to inspection. Indeed (107) at $s=0$ reads

$${\tau }^{\nu }\left(t\right)=c\nu t+1+\alpha ln(1-\alpha )-\alpha ln[{\tau }^{\nu }\left(t\right)-\alpha ],$$

an exact relation in which the coefficient of the logarithm is $\alpha =c_L/\nu$ and the additive constant is $\mathfrak{c}=1+\alpha ln(1-\alpha )$, that is $0.9280795$ for Model T. The coefficient is thus $c_L/\nu$ and not $L_{\infty }/\nu$, which for this model would be five times larger; see Remark 2.

10.2. The Three Regimes

For Model T one has ${\rho }_{*}=\delta$ and hence ${\rho }_1=\left|\gamma \right|$ in (62), and indeed the deficit of the last column of Table 3 below unity falls by the factor ${10}^{-{\rho }_1/\nu }=0.5623$ per decade, in agreement with the second-order rate. The convergence is slow because $\chi /\nu =1/2$; this is a feature of the regime and not of the estimate.

Figure 1 exhibits the assertion of Theorem 7 that the relative deviation $\zeta (t;s)$ is free of s to leading order, and is worth reading as a picture of the remainder in (62), in which two terms of the same relative order ${\left[\tau \left(t\right)\right]}^{-\left|\gamma \right|}$ compete. At $s=0$ the term $\mathcal{D}(t;s)$ of (66) vanishes identically, so that what is left is the second-order part of ${\mathcal{J}}_0\left(t\right)$, which is negative and holds the curve below unity; for $s>0$ the term $-\mathcal{D}(t;s)>0$ enters, proportional to $M\left(s\right)$, and for the two values shown it outweighs that deficit. Since $M\left(s\right)$ grows without bound as $s\uparrow 1$ while the order in t is the same, the excess is governed by s, and at $t={10}^7$ it is still $12.9\%$ when $s=0.9$. The uniformity of (61) on $[0,d]$ is therefore genuine but is attained slowly when d is close to 1, and it is for that reason that the theorem is not stated on $[0,1)$.

The remaining two regimes are treated in Figure 2. At $s=0$ one has $M\left(0\right)=0$, so that the constant of (75) reduces to $-{\mathsf{\Lambda }}_{\infty }[c_{\ell }-c_L]/\left({\nu }^3{L}_{\infty }\right)$. The left panel makes visible the point of Remark 8: across the threshold the rate is the same, while the exponent ${\mathsf{\Lambda }}_{\infty }$ of the normalisation ${\tau }^{{\mathsf{\Lambda }}_{\infty }}\left(t\right)$ is not. The right panel confirms in the same way the monotone approach of Theorem 11.

10.3. The Correction Term in $M\left(s\right)$

Finally we quantify Remark 3. What Figure 3 shows is that the omitted term is not a correction of smaller order: it is comparable with the leading term over the whole range and disappears only in the limit, where both sides are dominated by the singularity.

Figure 3. Model T. The ratio ${\mathsf{D}}_4\left(s\right)$ of the exact $M\left(s\right)$ of (17) to the reduced expression $[1/\mathcal{V}(1-s)-1/a_0]/\nu$, which omits the term $\mathcal{I}(1/(1-s\left)\right)$ of (41). It falls monotonically between the dark lines at $3/2$ and 1. Both endpoints are limiting values: at $s=0$ the two quantities vanish together and the ratio tends to $\nu a_0/|a_0+f^{\prime }\left(0\right)|=3/2$, while as $s\uparrow 1$ both diverge and the ratio tends to 1.

Figure 3. Model T. The ratio ${\mathsf{D}}_4\left(s\right)$ of the exact $M\left(s\right)$ of (17) to the reduced expression $[1/\mathcal{V}(1-s)-1/a_0]/\nu$, which omits the term $\mathcal{I}(1/(1-s\left)\right)$ of (41). It falls monotonically between the dark lines at $3/2$ and 1. Both endpoints are limiting values: at $s=0$ the two quantities vanish together and the ratio tends to $\nu a_0/|a_0+f^{\prime }\left(0\right)|=3/2$, while as $s\uparrow 1$ both diverge and the ratio tends to 1.

10.4. Transfer to the transition probabilities

The bounds of Section 9 were tested on Model T with $d=0.6$, through

$${\mathsf{D}}_5(t;j):=d^{j}t|{\mathsf{p}}_j\left(t\right)/{\mathsf{p}}_1\left(t\right)-v_j|\mathrm{and}{\mathsf{D}}_6\left(t\right):={\displaystyle \frac{t}{q^{\gamma }\left(t\right)}}\underset{\left|s\right|=d}{sup}|\mathsf{r}(t;s)/\mathsf{r}\left(s\right)-1|,$$

each a deviation divided by the order predicted for it, by (93) and by (98) respectively. The coefficients entering ${\mathsf{D}}_5(t;j)$ are extracted by Cauchy’s formula as discrete means over equally spaced points of the circle $\left|s\right|=d$, at which (107) and (108) are evaluated with the principal branches of the powers and of the logarithm, and the ratio in ${\mathsf{D}}_6\left(t\right)$ is obtained from (97) by quadrature along the segment from $F(t;0)$ to $F(t;s)$. In Table 4 the deviations settle by $t={10}^3$ and decrease down the column, so that the geometric factor $d^{-j}$ of (93) is not merely permissible but generous, and the uniform bound (95) multiplied by t converges to $2.2437$. That ${\mathsf{D}}_6\left(t\right)$ settles at a positive constant indicates that, for this model, the order of (98) is attained and is not merely an upper bound.

11. Concluding Remarks

We collect the points on which the present treatment departs from the existing one, and the questions it leaves open.

On the hypotheses. Placing the assumptions on the index function ${\epsilon }_L(*)$ rather than on the shift remainder $w_{\mu }(*)$ is not a matter of taste: the two are proportional with a factor that depends on the shift and is therefore not determined once the shift has been suppressed, as Lemma 2 and Remark 1 make precise, whereas the index function is the object the representation theorem provides and the one all the integrations actually use. How the axioms so obtained sit inside the subclass (7) in which this circle of problems is usually set is recorded where they are stated, and it is the two second-order rates admitted there, combined in ${\rho }_{*}$ of (6), that turn the rates of (62) and (76) into powers of t. Remark 2 records what is gained besides.

On the exact identities. Lemma 3 and Lemma 7 are identities, not asymptotic relations, and the sharpness of everything downstream comes from them. In particular the difference (33) of the identity at two values of s has no logarithmic term at all, the logarithms cancelling exactly; this is what makes the s-uniform statements (61), (75) and (45) possible, and what makes the s-dependent part of the correction of Theorem 4 computable: it is quadratic in $M\left(s\right)$ and comes from the second-order term of a single expansion, not from the logarithms.

On the transfer to the transition probabilities. The passage from a rate for a generating function to a rate for its coefficients is usually either omitted or bought with a monotonicity hypothesis. Section 9 shows that in the present model it is free. Criticality factors the infinitesimal generating function as $f\left(z\right)={(1-z)}^2\mathsf{\Omega }\left(z\right)$ with $\mathsf{\Omega }(*)$ a nonvanishing power series with nonnegative coefficients, and this single fact yields both the analyticity of $M(*)$ on the whole unit disc and, through Cauchy’s estimate on a disc of radius $q\left(t\right)/2$ around $F(t;0)$, the complex bounds of Lemma 10. Off the real axis the argument uses no property of the branching system beyond criticality and the nonnegativity of the rates, the regular variation entering only through the real-variable estimates already proved, and it should apply verbatim to other critical models in which the generating function satisfies an autonomous first-order equation.

On the classification. Theorem 6 completes the sign criterion of [18,25] by settling the boundary $\gamma =0$, where the sign carries no information and the threshold is ${\mathsf{\Lambda }}_{\infty }=\nu$. It would be of interest to know whether the equality case ${\mathsf{\Lambda }}_{\infty }=\nu$ can be refined: there $p_{00}\left(t\right)$ is of exact order $1/t$ by (54), so that the chain is null recurrent with a return integral that diverges only logarithmically, but the tail of the return time to the origin is not addressed here.

Open question. One restriction remains. The transient regime is treated under the hypothesis that $\chi$ of (5) is positive, which is what makes the integral $\mathcal{K}\left(s\right)$ of (56) converge. For $\chi \le 0$ the limiting generating function (59) does not exist in the present form and a different normalisation is required; we do not know what it is. The other hypothesis of that regime, $c_{\ell }>0$, is not a restriction of the same kind: what replaces the rate (61) when it fails is, under $\left({\mathcal{B}}^{*}\right)$ with ${\rho }_{\ell }>\left|\gamma \right|$, the rate (68) of Remark 7, and the limit itself is unaffected in any case. Both are restrictions on the parameters, not on the method.