Convergence towards linear combinations of chi-squared random variables: a Malliavin-based approach

Abstract

We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators ${\Gamma }_i$ (introduced by Nourdin and Peccati in \cite{n-pe-3}), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly \cite{n-po-1}, concerning the limiting behaviour of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times.

References in corpus (6)

• Central limit theorems for sequences of multiple stochastic integrals
• Noncentral convergence of multiple integrals
• Chaos of a Markov operator and the fourth moment condition
• Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws
• Generalization of the Nualart-Peccati criterion
• Malliavin-Stein method for Variance-Gamma approximation on Wiener space

Cited by in corpus (1)

• A bound on the 2-Wasserstein distance between linear combinations of independent random variables