Almost Sure convergence of sum of independent random variables

Question

Let $\{{X_{j}}\}_{1}^{\infty}$ be independent r.v.s such that $\sum E( |X_{j}|) <\infty$. How to show that $\sum X_{j}$ converges almost surely. Can I argue simply that for every $\epsilon>0, \exists N$ such that $\forall j,k >N, E(|X{j}-X_{k}|)<\epsilon$. Then I proceed exactly as in

how to show convergence in probability imply convergence a.s. in this case?

can someone come up with a proof which does not involve Ottiavani's inequality..... — user24367, Feb 17 '13 at 01:13

passerby51 · Accepted Answer · 2013-02-17T05:05:12.683

11

Here is a simple proof. By monotone convergence theorem: $$ \sum_j E|X_j| = E \big[ \sum_{j} |X_j| \big]. $$ It follows from the assumption that $E \big[ \sum_j |X_j| \big] < \infty$. Any random variable which has finite expectation should be finite almost surely. Thus, $\sum_j |X_j| < \infty$ almost surely. But absolute convergence for series implies convergence, hence $\sum_j X_j$ converges almost surely.

edited Feb 17 '13 at 05:05

answered Feb 17 '13 at 01:44

passerby51

4,140

so ${X_{j}}$ need not be independent? – user24367 Feb 17 '13 at 04:35
It doesn't seem so. – passerby51 Feb 17 '13 at 04:48
in your argument where did you use the fact of independence – user24367 Feb 17 '13 at 04:52
I meant you don't need independence. – passerby51 Feb 17 '13 at 04:58
I have also made an edit. The last line should be $\sum_j X_j$ converges almost surely (not $\sum_j X_j < \infty$), since this is more precise. – passerby51 Feb 17 '13 at 05:06
This solves problem 7 on page 129 of Chung's "A Course in Probability Theory." – Jeremy Teitelbaum Jan 05 '17 at 21:50

Almost Sure convergence of sum of independent random variables

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