I have that $Y_1, Y_2, ..., Y_n$ are i.i.d. Poisson random variables with mean 1, and that $U_n = \sqrt{\frac{\sum_{i=1}^{n}{Y_i^2}}{n}}$.
Given that I have a sequence $U_1, U_2, ..., U_n$, I'm trying to find the pdf/pmf of the limiting distribution for $U$.
I believe that I can use the Weak Law of Large Numbers to handle $\sqrt{\frac{\sum_{i=1}^{n}{Y_i^2}}{n}}$, since that expression is comparable to $\overline{Y^2}$ (if that's a thing). I believe that I would be trying to take the limit of the MGF of $U$ to see if there a limiting distribution.
I tried using various approaches, such as letting $g(y) = y^2$, but I'm getting stuck with my math. Any pointers?