MGF Technique

Definition

Let $\{X_n\}$ be a Sequence of random variables with MGF $M_{X_n}(t)$ and $X$ be a Random Variable with MGF $M_X(t)$, then $\underset{n\to \infty }{\lim }{M}_{X_n}(t)=M_X(t)\iff X_n\stackrel{D}{\to }X$

A technique calculating limiting distribution using Moment Generating Function

Facts

$\underset{n\to \infty }{\lim }\psi (n)=0\Rightarrow \underset{n\to \infty }{\lim }(1+\frac{b}{n}+\frac{\psi (n)}{n}{)}^{cn}=e^{bc}$