Noncentral convergence of multiple integrals

Abstract

Fix $\nu >0$, denote by $G(\nu /2)$ a Gamma random variable with parameter $\nu /2$ and let $n\ge 2$ be a fixed even integer. Consider a sequence $\{F_k{\}}_{k\ge 1}$ of square integrable random variables belonging to the $n$th Wiener chaos of a given Gaussian process and with variance converging to $2\nu$. As $k\to \infty$, we prove that $F_k$ converges in distribution to $2G(\nu /2)-\nu$ if and only if $E(F_k^4)-12E(F_k^3)\to 12{\nu }^2-48\nu$.

Published in at http://dx.doi.org/10.1214/08-AOP435 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Central limit theorems for sequences of multiple stochastic integrals

Cited by in corpus (3)

• Central limit theorems for multiple Skorohod integrals
• Moments, cumulants and diagram formulae for non-linear functionals of random measures
• Asymptotic expansions for functions of the increments of certain Gaussian processes