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