# question on convergence

I'm working on a question I appreciate if you can guide me on how to solve it.

Consider $X_j$ as integer random variables.

We know $Pr(X_j = k) \rightarrow Pr(X = k)$ for every integer k. We also know that X is an integer-valued random variable.

I'd like to show:

$\sum_{k = -\infty}^{\infty} \max\{Pr(X = k) - Pr(X_j = k), 0\} \rightarrow 0$, as $j \rightarrow \infty$

Thanks for your help in advance.

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## 1 Answer

(Edited: My first answer was flawed, on closer inspection.)

Hint: The $k$th term in the given sum lies between $0$ and $Pr(X=k)$. The sum of the latter converges, so given $\varepsilon>0$, there is some $N$ with $$\sum_{\lvert k\rvert>N} Pr(X=k)<\epsilon.$$ Now you only need to take care of the remaining terms, which are finite in number.

Alternative, if you know your integration theory well enough to understand that a sum is just an integral with respect to counting measure: Use the dominated convergence theorem.

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Hi @Harald, Can you guide me a little more? –  user52144 Dec 12 '12 at 22:28
Sorry, my first hint was misleading. I mixed up the max and the sum somehow. I rewrote the answer. This time, I am sure it's right. –  Harald Hanche-Olsen Dec 12 '12 at 23:50