Convergence in distribution - Gamma distribution/degenerate distribution

Question

I got a gamma distribution which is defined as followed

$G_{n,\lambda}=\frac{\lambda e^{-\lambda x}(\lambda x)^{n-1}}{(n-1)!}$.

The paper I am currently reading says, that die $G_{n, n/t}$ converges in distribution to the degenerate distribution.

I don't know why this is true. Does someone has a hint or the answer for me ?

Thanks a lot !

Answer 1

The Moment Generating Function of your rv is

$$MGF_{X}(k)=(1-\frac{kt}{n})^{-n}$$

Its limit for $n\rightarrow +\infty$ is

$$e^{kt}$$

That is exactly the MGF of a degenerate rv