Noncentral convergence of multiple integrals

Fix $\nu >0$, denote by $G(\nu \mathrm{/}2)$ a Gamma random variable with parameter $\nu \mathrm{/}2$ 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 $k\to \mathrm{\infty }$, we prove that $F_k$ converges in distribution to $2G(\nu \mathrm{/}2)-\nu$ if and only if $E(F_k^4)-12E(F_k^3)\to 12{\nu }^2-48\nu$.