Noncentral convergence of multiple integrals

TL;DR

This paper establishes a precise criterion for the convergence in distribution of sequences of multiple Wiener chaos integrals to a shifted Gamma distribution, extending the understanding of noncentral limit theorems.

Contribution

It provides a necessary and sufficient condition involving moments for the convergence of multiple Wiener chaos integrals to a noncentral Gamma distribution.

Findings

Convergence in distribution occurs iff specific moment conditions are met.

The result characterizes noncentral limit behavior for multiple integrals.

The criterion applies to sequences with variance approaching a fixed value.

Abstract

Fix $\nu>0$, denote by $G(\nu/2)$ a Gamma random variable with parameter $\nu/2$ and let $n\geq2$ be a fixed even integer. Consider a sequence $\{F_k\}_{k\geq1}$ 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 $E(F_k^4)-12E(F_k^3)\to12\nu^2-48\nu$.