Chaos of a Markov operator and the fourth moment condition

Abstract

We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we are led to introduce a notion of chaos associated to a Markov operator through its iterated gradients and present conditions on the (pure) point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussian distribution. Convergence to gamma distributions may be examined similarly.

Published in at http://dx.doi.org/10.1214/11-AOP685 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Central limit theorems for sequences of multiple stochastic integrals
• Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos
• Noncentral convergence of multiple integrals

Cited by in corpus (19)

• A bound on the 2-Wasserstein distance between linear combinations of independent random variables
• Generalization of the Nualart-Peccati criterion
• Fourth moment theorems on the Poisson space in any dimension
• Fourth Moment Theorems for complex Gaussian approximation
• A Peccati-Tudor type theorem for Rademacher chaoses
• On the fourth moment theorem for the complex multiple Wiener-Itô integrals
• Entropy and the fourth moment phenomenon
• On the fourth moment condition for Rademacher chaos
• Superconvergence phenomenon in Wiener chaoses
• Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach
• Malliavin-Stein Method: a Survey of Recent Developments
• New moments criteria for convergence towards normal product/tetilla laws
• Stein's method, logarithmic Sobolev and transport inequalities
• On the Rate of Convergence to a Gamma Distribution on Wiener Space
• The fourth moment theorem on the Poisson space
• Convergence in total variation on Wiener chaos
• Malliavin calculus for marked binomial processes: portfolio optimisation in the trinomial model and compound Poisson approximation
• Stein-Malliavin Approximations for Nonlinear Functionals of Random Eigenfunctions on ${\mathbb{S}}^d$
• Fourth moment theorems on the Poisson space: analytic statements via product formulae