Fourth Moment Theorems for Markov diffusion generators

Abstract

A. Inspired by the insightful article [4], we revisit the Nualart-Peccati-criterion [13] (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.

Quantitative clts on a gaussian space: a survey of recent developments

“…in [5,44,49,52]. The paper [24] provides similar estimates for random variables living inside the chaos of a Markov operator; further important developments can be found in [2,3]. Analogous statements in the setting of free probability are discussed in [13,22].…”

“…[5,44,49,52]); the second one deals with limit theorems for stochastic integrals in the framework of free probability (see e.g. [13,22]); the third one deals with fourth moment theorems for random variables living in the chaos of a general Markov diffusion generator (see [2,3,24]). The reader can consult the constantly updated webpage http://www.iecn.u-nancy.fr/∼nourdin/steinmalliavin.htm for many applications of Malliavin-type techniques, as well as for asymptotic results that are somehow connected with the fourth moment theorem proved in [42].…”

Stein operators for variables form the third and fourth Wiener chaoses

“…The case n = 2 corresponds to the centered chisquare random variable Z 2 − 1. This is a special case of the centered gamma distribution, for which the Malliavin-Stein method is also highly tractable; see [3,4,12,28]. As with Stein's method for normal approximation, at the heart of Stein's method for H 2 (Z) = Z 2 − 1 is the Stein equation…”

Four moments theorems on Markov chaos

“…Building on [Led12], Azmoodeh et al showed in [ACP14] that the spectral condition can be replaced with a Markov chaos property of the eigenfunctions which is less restrictive than the earlier notion of Markov chaos introduced in [Led12]. In addition to Four Moments Theorems for convergence towards the Gaussian and Gamma distributions, a Four Moments Theorem for the approximation of the Beta distribution was proven.…”

“…It can be seen as an abstract version of the Malliavin-Stein method on Wiener chaos, first introduced in [NP09b]. Then, in order to further bound (3) by the right hand side of (2) when G is a chaotic element of the Markov structure and the law of Z belongs to the Pearson family, we again make use of the Gamma calculus and spectral arguments that, in a similar spirit as in [ACP14], allow us to obtain a linear combinations of the first four moments as a bound for the right hand side of (3). Note that, in general, one cannot a priori use moments to prove convergence towards a heavy-tailed distribution.…”