If $X_1,X_2......$ follow Poisson$(λ)$. Can we find suitable constants $a_n$ and $b_n$ such that $a_n(Y_n - b_n)$ converges to a non degenerate limit where $Y_n = (1 - \frac{1}{n})^{n\bar{X}_n}$.
I have shown that $Y_n\rightarrow e^{-\lambda}$ almost surely. However I cant find $a_n$ and $b_n$ such that the desired limit will be a non-degenerate distribution.