Let $(X_n)_{n \ge 0}$ be independent random variables, such that $X_n$ has Poisson distribution with parameter $n^2$. I want to find almost sure limit of the sequence $Y_n = \frac{X_1 \cdot X_2\cdot \ldots \cdot X_n}{(n!)^2}$.
It's easy to check, that $Y_n$ is a nonnegative martingale, so the almost sure limit does exist. To calculate it, I've tried changing the sequence to $e^{\sum \ln{(X_k/k^2)}}$ and using SLLN, but of course there is a problem with the value $0$. But still, I suspect that the limit is just equal to $0$.
Edit. Actually, I'm not that sure the limit will be $0$...