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Let $X_n$, $n = 1, 2, 3, \ldots$, and $X$ are random variables with at most countably many integer values. Prove that that $X_n \to X$ weakly if and only if $\lim_{n \to \infty} P (X_n = j) = P(X = j)$ for every $j$ in the state space of $X_n$, $n = 1, 2, 3, \ldots$, and $X$.

I managed to prove one direction, i.e. if $X_n$ converges to $X$ weakly than the requirement holds. Would be grateful if you could give an advice how to prove the other direction: assuming $\lim_{n \to \infty} P (X_n = j) = P(X = j)$ for every $j$ in the state space of $X_n$, $n = 1, 2, 3, \ldots$, and $X$, then $X_n \to X$ weakly.

Thanks.

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    Have the random variables _integer_ values? Otherwise, take $X_n=1/n$ and $X=0$: there is weak convergence, but $P(X_n=0)=0$ which doesn't converge to $P(X=0)=1$.2012-12-02
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    Yes, sorry didn't state it explicitly. The random variables take integer values. One can also state the problem a bit more generally: let $\mu_n$ and $\mu$ probability measures defined on some discrete space $X$ with countable many elements. Then $\mu_n \to \mu$ weakly iff for every $x \in X$ holds: $\mu_n(\{ x\}) \to \mu(\{ x \})$.2012-12-02
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    What is the definition of weak convergence you have? (a good exercise is to check it for all the (equivalent) definitions)2012-12-02
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    Definition. $\mu_n \to \mu$ weakly if $\int fd\mu_n \to \int fd\mu$ for every continuous bounded function on the underlying space $X$.2012-12-02
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    Note that if $X$ is discrete then *every* function on $X$ is continuous.2012-12-02

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Let $f$ a continuous bounded function, $p_{n,k}:=P(X_n=k)$ and $p_k:=P(X=k)$. If $f$ is continuous and bounded, then $$\int f(X_n)dP=\sum_{k\in\Bbb Z}p_{n,k}f(k)$$ (this makes sense as $f$ is bounded). Using boundedness, the result follows from dominated convergences with counting measure on $\Bbb Z$.

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    Thanks, Davide. I see, the key point I forgot here is the dominated convergence to conclude the series convergence.2012-12-02
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    May I ask f(X) is continuous on R, but the random variable is defined on a countable set, does this matter?2015-09-24