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 (2)

• Central limit theorems for sequences of multiple stochastic integrals
• Noncentral convergence of multiple integrals

Cited by in corpus (6)

• 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
• 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 ${\mathbb{S}}^d$