The problem is to estimate the distribution of product of two $\textit{discretized Gaussian}$ random variables with zero means. The discretized Gaussian means that the p.m.f. looks like
$D_s(x)=\rho_s(x)/\rho(\mathbb{Z}), x \in \mathbb{Z}$,
where $\rho_s(x)=exp(-x^2/s^2)$ is Gaussian subject to discrete support and $\rho(\mathbb{Z})$ is the normalization.
The question is: how to show that the product of two independant discrete Gaussians $X_1 \leftarrow D_{s_1}, X_2\leftarrow D_{s_2}$ is a subgaussian (namely, the tail decays exponentially and how fast it decays) except the argument that the resulting variable is bounded since the st.dev=$s_1s_2$? Assume no relatively good p.m.f. is expected like in continuous case (Normal Product Distribution)?
Thanks in advance.