Kim, Y.-T.; Park, H.-S. Improved Bound of Four Moment Theorem and Its Application to Orthogonal Polynomials Associated with Laws. Axioms2023, 12, 1092.
Kim, Y.-T.; Park, H.-S. Improved Bound of Four Moment Theorem and Its Application to Orthogonal Polynomials Associated with Laws. Axioms 2023, 12, 1092.
Kim, Y.-T.; Park, H.-S. Improved Bound of Four Moment Theorem and Its Application to Orthogonal Polynomials Associated with Laws. Axioms2023, 12, 1092.
Kim, Y.-T.; Park, H.-S. Improved Bound of Four Moment Theorem and Its Application to Orthogonal Polynomials Associated with Laws. Axioms 2023, 12, 1092.
Abstract
We introduce the lower chaos grade of a real-valued function F defined on the Markov triple (E,μ,Γ), where μ is a probability measure and Γ is the carré du champ operator. As an application of this concept, we obtain the better estimate of the four moments theorem for Markov diffusion generators worked by Bourguin et al (2019). For our purpose, we need to find the largest number except zero in the set of eigenvalues corresponding to its eigenfunction in the case where the square of a random variable F, coming from a Markov triple structure, can be expressed as a sum of eigenfunctions, We give some examples of eigenfunctions of the diffusion generators such as Ornstein-Uhlenbeck, Jacobi and Romanovski-Routh. In particular, two bounds, called the four moments theorem and fourth moment theorem respectively, will be provided for the normal approximation of the case where a random variable F comes from eigenfunctions of a Jacobi generator.
Keywords
Markov diffusioin generator; carré du champ operator; Pearson distribution; Fourth moment theorem; Malliavin calculus; Jacobi generator; Romanovski-Routh generator
Subject
Computer Science and Mathematics, Probability and Statistics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
