Let $Z_n \sim N(0, n^{-1})$ be independent. I would like to know what I can say about the distribution of
$$Z := \sum_{n = 1}^{\infty} Z_n^2$$
Each $Z_n^2$ can be written as $\frac{Y_n}{n}$,where the $Y_n$ are i.i.d. $\chi^2_1$. However, I am not sure how we could then get a closed form for the distribution of $Z$. Any insight would be appreciated