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?