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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.

user87455
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  • What if you consider the same problem with $Y_n = e^{-\overline{X}_n}$? Maybe if you can come up with constants for that, they may work in the original problem as well. – user2566092 Aug 29 '13 at 17:58
  • I'd try to show that variance of $Y_n$ decreases as $1/n$, and hence $a_n = 1/\sqrt{n}$, $b_n = e^{-\lambda}$ (or $b_n = (1-1/n)^{n\lambda}$ should work. – leonbloy Aug 30 '13 at 01:21
  • https://math.stackexchange.com/q/4107774/321264 – StubbornAtom Mar 07 '23 at 18:13

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