A Uniform Polynomial Rate for the Extreme-Value Law of ϑ-Expansion Digits

1. Introduction and Main Results

Fix $\vartheta \in (0,1]$ with

$${\vartheta }^2={\displaystyle \frac{1}{m}},m\in \mathbb{N},$$

and let $T_{\vartheta }:[0,\vartheta ]\to [0,\vartheta ]$ be the generalized Gauss map

$$T_{\vartheta }\left(x\right)={\displaystyle \frac{1}{x}}-\vartheta ⌊{\displaystyle \frac{1}{\vartheta x}}⌋(x\ne 0),T_{\vartheta }\left(0\right)=0.$$

Every irrational $x\in (0,\vartheta )$ has a $\vartheta$-expansion

$$x={\displaystyle \frac{1}{a_1\vartheta +\frac{1}{a_2\vartheta +\frac{1}{a_3\vartheta +\dots }}}},a_j=a_j\left(x\right)\in {\mathbb{N}}_m:=\{m,m+1,\dots \}.$$

The metrical theory of these expansions was initiated by Chakraborty and Rao [2]. In the case ${\vartheta }^2=1/m$ the transformation preserves the explicit probability measure

$$d{\gamma }_{\vartheta }\left(x\right)={\displaystyle \frac{\vartheta }{\log (1+{\vartheta }^2)}}{\displaystyle \frac{dx}{1+\vartheta x}},0\le x\le \vartheta ,$$

(1.1)

and the digit process ${\left(a_j\right)}_{j\ge 1}$ is strictly stationary. Sebe, Lascu and Selmi [8] further record an exponential $\psi$-mixing estimate for this process.

Put

$$M_n=\underset{1\le j\le n}{\max }{a}_j,L_{\vartheta }=\log (1+{\vartheta }^2),c_{\vartheta }={\displaystyle \frac{1}{L_{\vartheta }}}.$$

The extreme-value theory of continued-fraction digits goes back to Galambos [4] and Philipp [6]. Quantitative exponential and extreme- value approximations for dependent processes have also been developed in a broader mixing framework; see, for example, Abadi [1] for exponential hitting-time errors and Freitas, Freitas and Todd [3] for rates in rare-event and extreme-value laws. Quantitative Poisson approximation for extremes of dependent sequences was developed, in particular, through the Stein–Chen approach of Smith [9]. For regular continued fractions, quantitative rates were substantially sharpened by Ghosh, Kirsebom and Roy [5]. For background on the metrical and ergodic theory of $\vartheta$-continued fractions, see also Sebe and Lascu [7]. For $\vartheta$-expansions, Sebe, Lascu and Selmi [8] proved the corresponding Fréchet law and quantitative error estimates. In the normalization used here their limit law is

$${\gamma }_{\vartheta }(M_n\le ny)⟶\exp \left(-{\displaystyle \frac{c_{\vartheta }}{y}}\right),y>0.$$

(1.2)

The statement of [8, Theorem 4.6] gives a subpolynomial remainder, while the proof contains polynomial estimates on restricted level ranges of the form

$$y\ge {\left(\log n\right)}^{-\delta },0<\delta <1.$$

Accordingly, the point of the present note is not the existence of a polynomial estimate by itself. We address the logarithmic boundary at which the available restricted-range argument degenerates: our block comparison remains effective down to

$$y\asymp {\left(\log n\right)}^{-1},$$

where the corresponding Poisson parameter is already of order $\log n$. Monotonicity then controls all smaller levels, yielding a uniform polynomial Kolmogorov bound for all $y>0$.

The argument uses two simple features of the $\vartheta$-system. First, the marginal tail is computed exactly from (1.1). Second, distant blocks are multiplicatively close to independent at an exponential rate. Section 3 isolates the resulting logarithmic-window estimate for a general stationary sequence with exponential $\psi$-mixing. The abstract estimate is an elementary tool rather than an optimal general rare-event theorem; its role here is to stay uniform while the Poisson parameter grows like $\log n$.

We use blocking rather than directly transplanting the Chen–Stein scheme for regular continued fractions. In the form used in [5], that argument includes a conditional $b_3$ term associated with observations outside a two-sided neighbourhood. Rather than verify the corresponding conditional estimate for the $\vartheta$-system, we work only with the available past–future $\psi$-mixing bound.

Our main result is the following.

Theorem 1.1.

Let ${\vartheta }^2=1/m$, $m\in \mathbb{N}$. There exists $C_{\vartheta }<\infty$ such that, for all $n\ge 3$,

$$\underset{y>0}{\sup }\left|{\gamma }_{\vartheta }(M_n\le ny)-\exp \left(-{\displaystyle \frac{c_{\vartheta }}{y}}\right)\right|\le C_{\vartheta }{\displaystyle \frac{{\left(\log n\right)}^2}{\sqrt{n}}}.$$

(1.3)

Moreover, for every fixed $\delta >0$ there exists $C_{\vartheta ,\delta }<\infty$ such that

$$\underset{y\ge \delta }{\sup }\left|{\gamma }_{\vartheta }(M_n\le ny)-\exp \left(-{\displaystyle \frac{c_{\vartheta }}{y}}\right)\right|\le C_{\vartheta ,\delta }{\left({\displaystyle \frac{\log n}{n}}\right)}^{1/2}.$$

(1.4)

The first estimate is the contribution emphasized here: it is a single polynomial Kolmogorov bound valid on the full half-line. Indeed, if

$$G_{\vartheta }\left(y\right)=\left\{\begin{array}{cc}0,\hfill & y\le 0,\hfill \\ \exp (-c_{\vartheta }/y),\hfill & y>0,\hfill \end{array}\right.$$

then, since $M_n/n>0$, the left-hand side of (1.3) is exactly the Kolmogorov distance between the law of $M_n/n$ and the distribution function $G_{\vartheta }$. The second estimate is included as a sharper bound on fixed level ranges, but it is not used as a separate priority claim. The extra logarithmic factor in (1.3) comes from pushing the block comparison into levels for which the effective Poisson parameter is of order $\log n$.

The proof rests on comparing the no-exceedance probability with its Poisson surrogate. We write

$$\tau \left(u\right):={\gamma }_{\vartheta }(a_1>u),\lambda =n\tau \left(u\right),$$

and compare ${\gamma }_{\vartheta }(M_n\le u)$ with $e^{-\lambda }$. The exact tail formula also gives an immediate i.i.d. benchmark.

Corollary 1.1.

Let ${\left(a_j^{*}\right)}_{j\ge 1}$ be i.i.d. with the same marginal law as $a_1$ and put $M_n^{*}={\max }_{1\le j\le n}{a}_j^{*}$. For every fixed $\delta >0$,

$$\underset{y\ge \delta }{\sup }\left|\mathbb{P}(M_n^{*}\le ny)-\exp \left(-{\displaystyle \frac{c_{\vartheta }}{y}}\right)\right|=O_{\vartheta ,\delta }\left(n^{-1}\right).$$

In addition,

$$\mathbb{P}(M_n^{*}\le u)={(1-\tau \left(u\right))}^n=\exp \left(-n\tau \left(u\right)-{\displaystyle \frac{1}{2}}n\tau {\left(u\right)}^2+O\left(n\tau {\left(u\right)}^3\right)\right)$$

uniformly whenever $n\tau \left(u\right)=O\left(1\right)$.

• Organization. Section 2 recalls the exact marginal tail and the exponential mixing estimate. Section 3 gives an abstract quantitative blocking principle, including the logarithmic Poisson window needed later. Section 4 derives the global and locally uniform rates in Theorem 1.1 and the i.i.d. benchmark.

2. Preliminaries

For $k\in {\mathbb{N}}_m$, the rank-one cylinder is

$$C_k=\{x:a_1\left(x\right)=k\}=\left({\displaystyle \frac{1}{\vartheta (k+1)}},{\displaystyle \frac{1}{\vartheta k}}\right],$$

up to immaterial endpoint conventions. Consequently, for every real $u\ge m$,

$$\{a_1>u\}=\left(0,{\displaystyle \frac{1}{\vartheta (\lfloor u\rfloor +1)}}\right].$$

(2.1)

Integrating (1.1) gives the exact tail.

Lemma 2.1

(Exact marginal tail). For $u\ge m$,

$$\tau \left(u\right)={\displaystyle \frac{1}{L_{\vartheta }}}\log \left(1+{\displaystyle \frac{1}{\lfloor u\rfloor +1}}\right).$$

(2.2)

In particular,

$$\tau \left(u\right)={\displaystyle \frac{c_{\vartheta }}{u}}+O_{\vartheta }\left(u^{-2}\right),u\to \infty .$$

(2.3)

Proof. 

By (2.1) and (1.1),

$$\tau \left(u\right)={\displaystyle \frac{1}{L_{\vartheta }}}{\int }_0^{1/\left(\vartheta \right(\lfloor u\rfloor +1\left)\right)}{\displaystyle \frac{\vartheta dx}{1+\vartheta x}}={\displaystyle \frac{1}{L_{\vartheta }}}\log \left(1+{\displaystyle \frac{1}{\lfloor u\rfloor +1}}\right).$$

Since ${(\lfloor u\rfloor +1)}^{-1}=u^{-1}+O\left(u^{-2}\right)$ and $\log (1+z)=z+O\left(z^2\right)$ as $z\downarrow 0$, (2.3) follows. □

We use the exponential $\psi$-mixing estimate established for the stationary $\vartheta$-digit process in [8]: there exist constants $K_{\vartheta }<\infty$ and $q_{\vartheta }\in (0,1)$ such that, whenever

$$A\in \sigma (a_1,\dots ,a_k),B\in \sigma (a_{k+r},a_{k+r+1},\dots ),$$

one has

$$\left|{\gamma }_{\vartheta }(A\cap B)-{\gamma }_{\vartheta }\left(A\right){\gamma }_{\vartheta }\left(B\right)\right|\le K_{\vartheta }{q}_{\vartheta }^r{\gamma }_{\vartheta }\left(A\right){\gamma }_{\vartheta }\left(B\right).$$

(2.4)

In particular, by stationarity,

$${\gamma }_{\vartheta }(a_i>u,a_j>u)\le (1+K_{\vartheta })\tau {\left(u\right)}^2,i\ne j.$$

(2.5)

3. A Quantitative Blocking Principle

We first isolate the probabilistic estimate used in the application. Let ${\left(X_j\right)}_{j\ge 1}$ be a strictly stationary sequence on a probability space with law $\mathbb{P}$, and write

$$M_n^X=\underset{1\le j\le n}{\max }{X}_j,p\left(u\right)=\mathbb{P}(X_1>u),\lambda =np\left(u\right).$$

Assume that there exist constants $K<\infty$ and $q\in (0,1)$ such that, for all $k,r\ge 1$,

$$\left|\mathbb{P}(A\cap B)-\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)\right|\le K{q}^r\mathbb{P}\left(A\right)\mathbb{P}\left(B\right)$$

(3.1)

whenever

$$A\in \sigma (X_1,\dots ,X_k),B\in \sigma (X_{k+r},X_{k+r+1},\dots ).$$

Notice that (3.1) implies, for $i\ne j$,

$$\mathbb{P}(X_i>u,X_j>u)\le (1+K)p{\left(u\right)}^2.$$

(3.2)

Lemma 3.1

(One-block estimate). Under (3.1), there exists $C_K<\infty$ such that, for every $b\in \mathbb{N}$ and every level u,

$$\left|\mathbb{P}(M_b^X\le u)-{(1-p\left(u\right))}^b\right|\le C_K{b}^{2}p{\left(u\right)}^2.$$

(3.3)

Proof. 

Write $A_i=\{X_i>u\}$. Bonferroni’s inequalities and (3.2) give

$$bp\left(u\right)-C_K{b}^{2}p{\left(u\right)}^2\le \mathbb{P}\left({\displaystyle \bigcup _{i=1}^b}{A}_i\right)\le bp\left(u\right).$$

On the other hand, $1-{(1-p)}^b=bp+O\left(b^2{p}^2\right)$. Subtracting the two relations proves (3.3). □

For integers $b,g\ge 1$ with $b+g\le n$, set

$$s=⌊{\displaystyle \frac{n}{b+g}}⌋,r_j=(j-1)(b+g),$$

and define

$$I_j=\{r_j+1,\dots ,r_j+b\},1\le j\le s.$$

and put

$$B_j=\left\{\underset{i\in I_j}{\max }{X}_i\le u\right\}.$$

Thus two consecutive retained blocks are separated by exactly g deleted coordinates, and $B_j$ begins $g+1$ indices after the end of $I_{j-1}$.

Proposition 3.1.

(Block comparison). Assume (3.1). There exists $C_K<\infty$ such that, whenever $b+g\le n$ and $bp\left(u\right)\le 1/2$,

$$\begin{array}{cc}\hfill \left|\mathbb{P}(M_n^X\le u)-e^{-\lambda }\right|\le C_K(& \lambda {\displaystyle \frac{g}{b}}+\lambda {\displaystyle \frac{b+g}{n}}+{\displaystyle \frac{n{q}^g}{b}}\hfill \\ \hfill & +{\displaystyle \frac{b{\lambda }^2}{n}}+{\displaystyle \frac{{\lambda }^2}{n}}).\hfill \end{array}$$

(3.4)

Proof. 

Deleting the gaps and the final remainder changes the no-exceedance event only if an exceedance occurs at one of at most $sg+b+g$ deleted sites. Hence

$$0\le \mathbb{P}\left({\displaystyle \bigcap _{j=1}^s}{B}_j\right)-\mathbb{P}(M_n^X\le u)\le (sg+b+g)p\left(u\right)\le C\left(\lambda {\displaystyle \frac{g}{b}}+\lambda {\displaystyle \frac{b+g}{n}}\right).$$

(3.5)

To decouple the retained blocks, put

$$E_j={\displaystyle \bigcap _{\ell =1}^j}{B}_{\ell },D_j=\left|\mathbb{P}\left(E_j\right)-{\displaystyle \prod _{\ell =1}^j}\mathbb{P}\left(B_{\ell }\right)\right|.$$

If k is the last index of $I_{j-1}$, then $B_j$ starts at $k+g+1$. Therefore (3.1), with $r=g+1$, gives

$$\left|\mathbb{P}(E_{j-1}\cap B_j)-\mathbb{P}\left(E_{j-1}\right)\mathbb{P}\left(B_j\right)\right|\le K{q}^{g+1}\mathbb{P}\left(E_{j-1}\right)\mathbb{P}\left(B_j\right)\le K{q}^g.$$

Consequently,

$$D_j\le D_{j-1}\mathbb{P}\left(B_j\right)+K{q}^g\le D_{j-1}+K{q}^g.$$

Since $D_1=0$, induction yields $D_s\le (s-1)K{q}^g$, and hence

$$\left|\mathbb{P}\left({\displaystyle \bigcap _{j=1}^s}{B}_j\right)-{\displaystyle \prod _{j=1}^s}\mathbb{P}\left(B_j\right)\right|\le C_K{\displaystyle \frac{n{q}^g}{b}}.$$

(3.6)

By stationarity and Lemma 3.1,

$$\left|{\displaystyle \prod _{j=1}^s}\mathbb{P}\left(B_j\right)-{(1-p\left(u\right))}^{sb}\right|\le C_{K}s{b}^{2}p{\left(u\right)}^2\le C_K{\displaystyle \frac{b{\lambda }^2}{n}}.$$

(3.7)

Since $0\le n-sb\le sg+b+g$, a telescoping estimate gives

$$\left|{(1-p\left(u\right))}^{sb}-{(1-p\left(u\right))}^n\right|\le C\left(\lambda {\displaystyle \frac{g}{b}}+\lambda {\displaystyle \frac{b+g}{n}}\right).$$

(3.8)

Finally, $bp\left(u\right)\le 1/2$ implies $p\left(u\right)\le 1/2$, and therefore

$$\left|{(1-p\left(u\right))}^n-e^{-np\left(u\right)}\right|\le Cnp{\left(u\right)}^2=C{\displaystyle \frac{{\lambda }^2}{n}}.$$

(3.9)

Combining (3.5)–(3.9) proves (3.4). □

Corollary 3.1.

(Logarithmic-window estimate). Assume (3.1). For every $A>0$ there exist constants $C_{A,K,q}<\infty$ and $n_0=n_0(A,K,q)$ such that, for all $n\ge n_0$ and all levels u satisfying

$$\lambda =np\left(u\right)\le A\log n,$$

one has

$$\left|\mathbb{P}(M_n^X\le u)-e^{-np\left(u\right)}\right|\le C_{A,K,q}{\displaystyle \frac{{\left(\log n\right)}^2}{\sqrt{n}}}.$$

(3.10)

Proof. 

Choose $B>3/\left|\log q\right|$ and set

$$b=\lfloor \sqrt{n}\rfloor ,g=\lceil B\log n\rceil .$$

For all sufficiently large n, $b+g\le n$ and

$$bp\left(u\right)={\displaystyle \frac{b\lambda }{n}}\le A{\displaystyle \frac{\log n}{\sqrt{n}}}\le {\displaystyle \frac{1}{2}},q^g\le n^{-3}.$$

The five terms on the right-hand side of (3.4) are then, respectively,

$$O\left({\displaystyle \frac{{\left(\log n\right)}^2}{\sqrt{n}}}\right),O\left({\displaystyle \frac{\log n}{\sqrt{n}}}+{\displaystyle \frac{{\left(\log n\right)}^2}{n}}\right),O\left(n^{-5/2}\right),$$

and

$$O\left({\displaystyle \frac{{\left(\log n\right)}^2}{\sqrt{n}}}\right),O\left({\displaystyle \frac{{\left(\log n\right)}^2}{n}}\right).$$

Proposition 3.1 gives (3.10). □

Remark 3.1

(Relation with Chen–Stein). For regular continued fractions, Ghosh, Kirsebom and Roy [5] obtained a sharper quantitative Poisson approximation by the Chen–Stein method. In the form used there, a direct application in the present setting would require a conditional $b_3$ term associated with observations outside a two-sided neighbourhood. Rather than verify the corresponding conditional estimate for the ϑ-digit process, we use the blocking argument above, whose input is exactly the past–future multiplicative estimate (2.4). Corollary 3.1 is sufficient for the logarithmic Poisson window needed below.

4. Proofs of the Main Results

Proof of Theorem 1.1.

We apply Section 3 with $X_j=a_j$, $\mathbb{P}={\gamma }_{\vartheta }$, $p\left(u\right)=\tau \left(u\right)$, $K=K_{\vartheta }$ and $q=q_{\vartheta }$. Choose a constant $A_{\vartheta }>0$ so large that

$$q_{\vartheta }^{\lceil A_{\vartheta }\log n\rceil }\le n^{-3},n\ge 3,$$

and put

$$g_n=\lceil A_{\vartheta }\log n\rceil .$$

Locally uniform bound. Fix $\delta >0$ and let $y\ge \delta$, $u=ny$. By Lemma 2.1,

$$\lambda =n\tau \left(ny\right)\le C_{\vartheta ,\delta }$$

for all sufficiently large n. Take

$$b_n=⌊\sqrt{n{g}_n}⌋.$$

Then $b_n\tau \left(ny\right)=O_{\vartheta ,\delta }\left(\sqrt{g_n/n}\right)$, so the hypothesis of Proposition 3.1 holds for large n. Substitution in (3.4) yields

$$\underset{y\ge \delta }{\sup }\left|{\gamma }_{\vartheta }(M_n\le ny)-e^{-n\tau \left(ny\right)}\right|\le C_{\vartheta ,\delta }\sqrt{{\displaystyle \frac{\log n}{n}}}.$$

(4.1)

Moreover, (2.3) gives uniformly for $y\ge \delta$,

$$\left|n\tau \left(ny\right)-{\displaystyle \frac{c_{\vartheta }}{y}}\right|\le {\displaystyle \frac{C_{\vartheta ,\delta }}{n}}.$$

Since $|e^{-x}-e^{-z}|\le |x-z|$ for $x,z\ge 0$, combining this estimate with (4.1) proves (1.4).

Global bound. Set

$${\delta }_n={\displaystyle \frac{c_{\vartheta }}{4\log n}}$$

for all sufficiently large n. For $y\ge {\delta }_n$, Lemma 2.1 gives

$$\lambda =n\tau \left(ny\right)\le C_{\vartheta }\log n.$$

Hence Corollary 3.1, applied uniformly over these levels, yields

$$\underset{y\ge {\delta }_n}{\sup }\left|{\gamma }_{\vartheta }(M_n\le ny)-e^{-n\tau \left(ny\right)}\right|\le C_{\vartheta }{\displaystyle \frac{{\left(\log n\right)}^2}{\sqrt{n}}}.$$

(4.2)

Also, uniformly for $y\ge {\delta }_n$,

$$\left|n\tau \left(ny\right)-{\displaystyle \frac{c_{\vartheta }}{y}}\right|\le {\displaystyle \frac{C_{\vartheta }}{n{y}^2}}\le C_{\vartheta }{\displaystyle \frac{{\left(\log n\right)}^2}{n}},$$

so (4.2) gives the required bound on this range.

It remains to consider $0<y<{\delta }_n$. By monotonicity,

$${\gamma }_{\vartheta }(M_n\le ny)\le {\gamma }_{\vartheta }(M_n\le n{\delta }_n).$$

At $y={\delta }_n$ we have $c_{\vartheta }/y=4\log n$, while Lemma 2.1 gives $n\tau \left(n{\delta }_n\right)=4\log n+O({\left(\log n\right)}^2/n)$. Hence

$$e^{-n\tau \left(n{\delta }_n\right)}=n^{-4}\exp \left(O_{\vartheta }\left({\displaystyle \frac{{\left(\log n\right)}^2}{n}}\right)\right)\le C_{\vartheta }{n}^{-4}$$

for all sufficiently large n. Evaluating (4.2) at ${\delta }_n$ therefore gives

$${\gamma }_{\vartheta }(M_n\le n{\delta }_n)\le C_{\vartheta }{n}^{-4}+C_{\vartheta }{\displaystyle \frac{{\left(\log n\right)}^2}{\sqrt{n}}}.$$

Furthermore,

$$\exp \left(-{\displaystyle \frac{c_{\vartheta }}{y}}\right)\le \exp \left(-{\displaystyle \frac{c_{\vartheta }}{{\delta }_n}}\right)=n^{-4}.$$

Thus the same $O_{\vartheta }({\left(\log n\right)}^2/\sqrt{n})$ bound holds for $0<y<{\delta }_n$. Enlarging the constant to cover the finitely many remaining n proves (1.3). □

Proof of Corollary 1.1.

For the i.i.d. model,

$$\mathbb{P}(M_n^{*}\le u)={(1-\tau \left(u\right))}^n.$$

If $y\ge \delta$, then $\tau \left(ny\right)=O_{\vartheta ,\delta }(1/n)$ and hence

$${(1-\tau \left(ny\right))}^n=e^{-n\tau \left(ny\right)}\left(1+O_{\vartheta ,\delta }\left(n^{-1}\right)\right).$$

Lemma 2.1 also gives $n\tau \left(ny\right)=c_{\vartheta }/y+O_{\vartheta ,\delta }\left(n^{-1}\right)$, proving the first assertion. The second follows from the Taylor expansion $\log (1-z)=-z-z^2/2+O\left(z^3\right)$, uniformly when z is sufficiently small. □

Remark 4.1

(On possible refinements). The block method is not expected to be optimal. A sharper treatment could be based on a Chen–Stein decomposition if one establishes a uniform conditional estimate strong enough to control the corresponding $b_3$ term for the ϑ-digit exceedance indicators. The general Stein–Chen extreme-value framework of Smith [9] and the regular-continued-fraction analysis in [5] indicate the potential gain from such an approach. We do not pursue that additional conditional-mixing input here.