$X_n$ ~ $\Gamma(n,n)$, find the limit in Law of $X_n$

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?

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