Noncentral convergence of multiple integrals

Fix ν>0, denote by $G(ν/2)$ a Gamma random variable with parameter $ν/2$ 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 $2ν$. As $k\to\infty$, we prove that $F_k$ converges in distribution to $2G(ν/2)-ν$ if and only if $E(F_k^4)-12E(F_k^3)\to12ν^2-48ν$.