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How does one show

$$\lim_{\alpha \to \infty} e^{-t\sqrt{\alpha}}\left(1-\frac{t}{\sqrt{\alpha}}\right)^{-\alpha} = e^{t^2 / 2}?$$

Not homework, this is from this proof that the gamma distribution has a limiting distribution of the standard normal as $\alpha \to \infty$. It suggests using numerical techniques to find the limit of the above, but I would like to know if there is a good way to solve this manually. I tried L'Hopital's rule but got $e^t$, which is obviously incorrect.

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marked as duplicate by Guy Fsone, Claude Leibovici calculus Nov 11 '17 at 6:07

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  • $\begingroup$ +1 this looks like a limit worth scratching your head about! $\endgroup$ – Zach466920 Jun 24 '15 at 23:50
  • $\begingroup$ Duplicate of this and this. $\endgroup$ – Lucian Jun 25 '15 at 0:40
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Start with the exp/log trick, your limit is

$$\exp\left(\lim_{\alpha\to\infty}-t\sqrt\alpha+\alpha\log\left(1+{-t\over\sqrt\alpha}\right)\right).$$

Now let $\beta=\alpha^{-1}$.

$$\exp\left(\lim_{\beta\to 0^+}-t\beta^{-1/2}-\beta^{-1}\log\left(1+(-t\sqrt\beta)\right)\right).$$

Now here we can use the Maclaurin series for $\log(1+x)$ since we're going to $0$. We get

$$\exp\left(\lim_{\beta\to 0^+}-t\sqrt\beta^{-1/2}-\beta^{-1}(-t\sqrt\beta-{t^2\beta\over 2}+O(\beta^{3/2})\right).$$

The only term which doesn't vanish is ${t^2\over 2}$ so we recover the result.

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  • $\begingroup$ Slick! I wish I came to this myself but thank you. $\endgroup$ – mathjacks Jun 25 '15 at 1:27
  • $\begingroup$ @mathjacks glad to help. It was a fun problem, thanks for posting! $\endgroup$ – Adam Hughes Jun 25 '15 at 1:30

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