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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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    Note that if $X$ is discrete then *every* function on $X$ is continuous.2012-12-02

1 Answers 1

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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). Now the key point here is that $\sum_{k\in\Bbb Z}p_{n,k}=\sum_{k\in\Bbb Z}p_{k}=1$. More explicitely, fix a integer $R$. Denote $I_R:=\{k\in\mathbb Z, \left\lvert k\right\rvert \leqslant k\}$ and $J_R:=\{k\in\mathbb Z, \left\lvert k\right\rvert \gt k\}$. Then $$\tag{*}\left\lvert \sum_{k\in\Bbb Z}p_{n,k}f(k)-\sum_{k\in\Bbb Z}p_{k}f(k)\right\rvert\leqslant \left\lvert \sum_{k\in I_R}p_{n,k}f(k)-\sum_{k\in I_R}p_{k}f(k)\right\rvert+\left\lvert \sum_{k\in J_R}p_{n,k}f(k)-\sum_{k\in J_R}p_{k}f(k)\right\rvert.$$ We control the last term in the following way: \begin{align} \left\lvert \sum_{k\in J_R}p_{n,k}f(k)-\sum_{k\in J_R}p_{k}f(k)\right\rvert&=\left\lvert \sum_{k\in J_R}\left(p_{n,k}-p_k\right)f(k) \right\rvert\\ &\leqslant \sum_{k\in J_R}\left\lvert p_{n,k}-p_k \right\rvert \lvert f(k)\rvert\\ &\leqslant \lVert f\rVert_\infty \sum_{k\in J_R} p_{n,k} +\lVert f\rVert_\infty \sum_{k\in J_R} p_{n,k} \\ &= \lVert f\rVert_\infty \left(1-\sum_{k\in I_R} p_{n,k} \right)+\lVert f\rVert_\infty \left(1-\sum_{k\in I_R} p_{k} \right). \end{align} Plugging this estimate into (*) and taking $\limsup_{n\to+\infty}$ gives $$ \limsup_{n\to+\infty}\left\lvert \sum_{k\in\Bbb Z}p_{n,k}f(k)-\sum_{k\in\Bbb Z}p_{k}f(k)\right\rvert\leqslant 2\lVert f\rVert_\infty \left(1-\sum_{k\in I_R} p_{k} \right) $$ and since $R$ is arbitrary, we get the result.

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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