I am investigating the following idea. Let $X_n$ ~ $\Gamma(n,n)$. I want to find the limit in Law of this random variable.

I tried using Paul Levy theorem that says the following: If I find the limit of the characteristic function equal to some function $\theta(t)$ continuous in 0, then there exists a random variable X such as $X_n \to X$ with $\theta$ as its characteristic function.

Following this idea I tried calculating the following limit:

$$ \lim \Big(\frac{1}{1-int}\Big)^n$$ However to me this goes to 0. But that is not possible because such a characteristic function cannot exist. Is this the right approach?

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    $\begingroup$ There is another parameterization of $\Gamma$ distribution for which $$ \varphi_n=(1-it/n)^n\to e^{-it}. $$ $\endgroup$ – d.k.o. Jan 5 at 23:36
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    $\begingroup$ Indeed: scale $n$ or $1/n$? $\endgroup$ – Did Jan 6 at 0:34
  • $\begingroup$ @Did What do you mean by scale? Anyway, such a characteristic function is only for the X = 1 constant random variable, am I right? $\endgroup$ – qcc101 Jan 6 at 7:19
  • $\begingroup$ "What do you mean by scale?" One of the parameters of every gamma distribution. Please se the WP page, if refreshing your memories is needed. $\endgroup$ – Did Jan 6 at 9:42
  • $\begingroup$ I see. I used the following: $ \Big( 1 - \frac{it}{n} \Big)^{-n}$, but I get $e^{it}$ $\endgroup$ – qcc101 Jan 6 at 11:30

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