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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.

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    How do you define convergence in distribution? Which ways of proving it do you know?2012-11-28
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    Hi @did, I'm very new to this concept. I don't know how many different ways exist. The one I know is to show that any for any continuous function f: R $\rightarrow R$, $f(X_n) \rightarrow f(X_{\infty})$2012-11-28
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    That is not the correct definition of weak convergence. For every continuous bounded $f$, you need $E_{X_n}[f]\rightarrow E_{X_\infty}[f]$ where $E_{X_n}$ is the expectation with respect to the empirical distribution of $X_n$.2012-11-28
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    For an integer-valued random variable, you could define weak convergence as $Pr(X_n = m) \rightarrow Pr(X_{\infty} = m)$. Otherwise the Portmanteau theorem shows the definition of convergence of $E_{X_n}[f]\rightarrow E_{X_\infty}[f]$ is equivalent for an integer-valued variable to $Pr(X_n = m) \rightarrow Pr(X_{\infty} = m)$ (a consequence of theorem 29.1 in Billingsley).2012-11-28

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