Suppose given two $n$-degree polynomials $p_1$ and $p_2$. Each coefficient of $p_1$ and $p_2$ is independently sampled from Gaussian distribution with mean 0 and standard deviation $\sigma$, i.e. we sample $n$ times from Gaussian distribution and these sampled values are coefficients of $p_1$. Same goes to $p_2$.
I want to know the distribution of $p$=$p_1\cdot$$p_2$. Does $p$ comply with Gaussian distribution? If so, what is the standard deviation? Are there proofs for this question?
Thank you very much.