91Ó°ÊÓ

\({X_1},...{X_n}\) is a sequence of positive integer valued random variables.

Let \({F_n}\) be the cdf of \({X_n}\).

We have to show that \(\mathop {\lim }\limits_{n \to \infty } {F_n}\left( x \right) = F\left( x \right)\) for every point at which F is continuous.

Since F is the cdf of the integer-valued distribution, the continuity points are all non-integer values of x together with those integer values of x to which F assigns probability 0.

It is clear that it sufficient to prove that\(\mathop {\lim }\limits_{n \to \infty } {F_n}\left( x \right) = F\left( x \right)\)for every non-interger x, because continuity of F from the right and the fact that F is non decreasing will take care of the integers with zero probability.

For each non-integer x, let \({m_x}\) be the largest integer such that m<x. Then

\(\begin{align}{F_n}\left( x \right) &= \sum\limits_{k = 1}^m {{\rm P}\left( {{X_n} = k} \right)} \\ &= \sum\limits_{k = 1}^m {f\left( k \right)} \end{align}\)

\(\begin{align}{F_n}\left( x \right) &= F\left( m \right)\\ &= F\left( x \right)\end{align}\)

Hence \({X_n}\) converges in distribution to F, where the converges follows because the sums are finite.