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Can someone give me a sequence of independent random variables (or an example of it, with explanation, if possible) with mean 0 such that: $\frac{1}{n} \sum X_i \rightarrow - \infty $

Thank you.

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I must be getting something wrong, but isn't it that the left hand side of the limit is the mean of those variables? –  Mario Apr 9 '13 at 12:58
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@Mario: He said the random variables were independent. Not iid. Otherwise SLLN gives 0. –  Gautam Shenoy Apr 9 '13 at 13:42
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For sake of completeness, you should include the type of convergence you expect. –  Davide Giraudo Apr 9 '13 at 20:00
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1 Answer 1

Assume that $P[X_n=n^3-n]=1/n^2$ and $P[X_n=-n]=1-1/n^2$ for every $n$, then $E[X_n]=0$ for every $n$ and the series $\sum\limits_nP[X_n\ne-n]$ converges hence by (the easy part of) Borel-Cantelli lemma, $X_n=-n$ for every $n$ large enough, almost surely, in particular $\sum\limits_{i=1}^nX_i\sim-\frac12n^2$ almost surely, which is enough to conclude. The independence hypothesis is not needed.

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Yeah, and in fact: $\sum\limits_{i=1}^nX_i\sim-\frac12n^2 \rightarrow - \infty $ almost surely (for n $\rightarrow \infty$) –  JohnD Apr 12 '13 at 12:53
    
Thanks a lot :) –  JohnD Apr 12 '13 at 12:54
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