Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials

Yoon Tae Kim,Hyun Suk Park,Shanhe Wu

doi:10.1155/2021/9408651

Abstract

In this paper, we consider a quantitative fourth moment theorem for functions (random variables) defined on the Markov tripleE,μ,Γ, whereμis a probability measure andΓis the carré du champ operator. A new technique is developed to derive the fourth moment bound in a normal approximation on the random variable of a general Markov diffusion generator, not necessarily belonging to a fixed eigenspace, while previous works deal with only random variables to belong to a fixed eigenspace. As this technique will be applied to the works studied by Bourguin et al. (2019), we obtain the new result in the case where the chaos grade of an eigenfunction of Markov diffusion generator is greater than two. Also, we introduce the chaos grade of a new notion, called the lower chaos grade, to find a better estimate than the previous one.

Highlights

The aim of this paper is to find the fourth moment bound in the normal approximation of a random variable related to a general Markov diffusion generator
We find a bound of the form dKolð for the Kolmogorov distance between a stand Gaussian random variable Z and a random variable F defined on ðE, μ, ΓÞ related to a Markov diffusion generator L with an invariant measure μ
Applying the technique developed in this paper can eliminate the second term in (5), which means that, in such a random variable G, the fourth moment theorem holds unlike the previous result in [16], where the upper bound (5) for a sequence fFng of chaotic random variables is given in (111) of Remark 12 (iii) In this paper, another notion of chaos grade, called a lower chaos grade, is introduced and used to provide a better estimate than the previous one obtained from (5) in [16]

Summary

Introduction

The aim of this paper is to find the fourth moment bound in the normal approximation of a random variable related to a general Markov diffusion generator. We prove that the right-hand side of (7) can be represented as the sum of two integrals with the operators mentioned above, and use this representation to prove that the fourth moment bound (6) holds This is the innovation point of this work (ii) If the upper chaos grade of F is strictly greater than two, the constant ξ2 and QðGÞ in the bound (5) are given as follows: ξ2 > 0 and QðGÞ = VarðG2Þ. Applying the technique developed in this paper can eliminate the second term in (5), which means that, in such a random variable G, the fourth moment theorem holds unlike the previous result in [16], where the upper bound (5) for a sequence fFng of chaotic random variables is given in (111) of Remark 12.

Preliminaries

Chaos Grade and Orthogonal Polynomials

Fourth Moment Theorem

Markov Chaos

Conclusions