13 - Continuity Theorem for Moment Generating Functions

Continuity Theorem for Moment Generating Functions

Let $Y$ and $X_1,X_2,\cdots$ be random variables with mgfs $M_Y$ and $M_{X_1},M_{X_2},\cdots$
which are finite on $[-t_0,t_0]$ for some $t_0>0$

If

$$M_{X_n}(t)\to M_Y(t)\text{ as }n\to \infty \text{ for all }t\in [-t_0,t_0]$$

Then

$$X_n\stackrel{d}{\to }Y\text{ as }n\to \infty$$