Noncentral convergence of multiple integrals

Details Noncentral convergence of multiple integrals Noncentral convergence of multiple integrals

2007-09-25

arxiv.org/abs/0709.3903v3

Annals of Probability 2009, Vol. 37, No. 4, 1412-1426

Mathematics

Probability

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

Scientific paper

10.1214/08-AOP435

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 $E(F_k^4)-12E(F_k^3)\to12\nu^2-48\nu$.

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