Proof of convergence in distribution of a discrete random variable

I'm working on a question on "convergence in distribution" and I appreciate if you could guide me on how to approach this question:

Here is the question:

Let $X_n$ be integer-valued random variables. Show that $X_n \stackrel w{\longrightarrow} X_{\infty}$ converges in distribution if and only if $\mathrm{Pr}(X_n = m) \rightarrow \mathrm{Pr}(X_{\infty} = m)$ for each $m$.

I appreciate your help.

Solution 1:

For the necessity, by Portmanteau theorem (theorem 29.1 in Billingsley or theorem 3.2.21 in Dembo's Notes), for any open interval $(a,b)$ which contains no integer, $$P(a<X_\infty<b) \le \liminf_n P(a<X_n<b) = 0$$ which implies that $$P(a<X_\infty<b)=0$$ and further that, for any non-integer $c$, $$P(X_\infty=c)=0$$ since $P(X_\infty=c) \le \lim_{\epsilon \to 0} P(c-\epsilon<X_n<c+\epsilon) = 0$.

Using the above two facts, for any integer $k$, we have \begin{align} P(X_\infty=k) & = P(k-0.5<X_\infty<k+0.5) \\ & \le \liminf_n P(k-0.5<X_n<k+0.5) \\ & = \liminf_n P(X_n=k) \\ & \le \limsup_n P(X_n=k) \\ & = \limsup_n P(k-0.5 \le X_n \le k+0.5) \\ & \le P(k-0.5 \le X_\infty \le k+0.5) \\ & = P(X_\infty=k)\\ \end{align} which implies that $P(X_\infty=k) = \lim_n P(X_n=k)$

For sufficiency, you should assume $X_\infty$ is also an integer valued random variable and then check the point-wise convergence of the distribution functions $P(X_n \le x)$at continuous points of $P(X_\infty \le x)$.

If $X_\infty$ is not integer valued, the converse direction doesn't hold. Here is a counter-example. $P(X_n=k)=1/n$ for $k=1,2,..n$ such that for any $k$, $\lim_{n\to \infty} P(X_n=k)=0$. Let $X_\infty \sim N(0,1)$. Then $P(X_\infty=k)=0$. However, $X_n$ do not converge weakly to $N(0,1)$.