# Generating Functions and Limit Theorems

It is a fact which we will not try to prove that if $$X, X_1, X_2, \cdots$$, are random variables with moment generating functions $$M(s), M_1(s), M_2(s), \cdots$$ respectively then if there is an $$R>0$$ so that for all $$s \in [0,R]$$

$M_n(s) \longrightarrow M(s) \qquad n \rightarrow \infty$

then for all $$y$$

$\mathbf{P}( X_n < y) \longrightarrow \mathbf{P}( X < y) \qquad n \rightarrow \infty$

Similarly, if the all of the random variables have a range of the  non-negative integers and respectively have probability generating functions $$G(s), G_1(s), G_2(s), \cdots$$ then if for all $$s \in [0,1)$$

$G_n(s) \longrightarrow G(s) \qquad n \rightarrow \infty$

thenfor all $$k$$

$\mathbf{P}( X_n = k) \longrightarrow \mathbf{P}( X = k) \qquad n \rightarrow \infty$

These results can be used to prove limit theorems.