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.