Four moments theorems on Markov chaos

Abstract

A. We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.2010 Mathematics Subject Classification. 60F05, 60J35, 60J99.

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

“…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.…”

“…The authors in [15] introduce a Markov chaos of eigenfunctions being less restrictive than the notion of Markov chaos defined in [14] and also obtain the quantitative four moment theorem for convergence of the eigenfunctions towards Gaussian, gamma, and beta distributions. Furthermore, Bourguin et al in [16] prove that convergence of the elements of a Markov chaos to a Pearson distribution can be still bounded with just the first four moments of the form…”

“…where d is a suitable distance, Z is a random variable with the law belonging to the Pearson family, and G is a chaotic random variable defined on ðE, μÞ. Here, P and Q in (5) are polynomials of degree four, and the constants ξ 1 and ξ 2 are determined in terms of a chaos grade defined in Definition 3.5 of [16].…”

“…(i) Compared to previous works [14][15][16], our studies are not limited to an element belonging to a fixed eigenspace of Markov chaos. Our result is a remarkable extension in comparison with other works dealing with only random variables in a fixed eigenspace.…”

“…where L −1 is the pseudo-inverse of the underlying Markov generatorL and Γ is the carré du champ operator, which is the result of the study in [16]. However, in order to find the fourth moment bound (6), we introduce a new technique relying on two types of the operators given in [17,18].…”