  In the framework of classical probability, we consider the normal product
distribution $F_\infty \sim N_1 \times N_2$ where $N_1, N_2$ are two
independent standard normal random variable, and in the setting of free
probability, $F_\infty \sim \left( S_1 S_2 + S_2 S_1 \right)/\sqrt{2}$ known as
{\it tetilla law} \cite{d-n}, where $S_1, S_2$ are freely independent
normalized semicircular random variables. We provide novel characterization of
$F_\infty$ within the second Wiener (Wigner) chaos. More precisely, we show
that for any generic element $F$ in the second Wiener (Wigner) chaos with
variance one the laws of $F$ and $F_\infty$ match if and only if $\mu_4 (F)= 9
\, (\mbox{resp. }\varphi(F^4)=2.5)$, and $\mu_{2r}(F)= ((2r-1)!!)^2 \,
(\mbox{resp. }\varphi(F^{2r})=\varphi(F^{2r}_\infty ))$ for some $r \ge 3$,
where $\mu_r (F)$ stands for the $r$th moment of the random variable $F$, and
$\varphi$ is the relevant tracial state. We use our moments characterization to
study the non central limit theorems within the second Wiener (Wigner) chaos
and the target random variable $F_\infty$. Our results generalize the findings
in Nourdin \& Poly \cite{n-p-2w}, Azmoodeh, et. al \cite{a-p-p} in the
classical probability, and of Deya \& Nourdin \cite{d-n} in the free
probability setting.
