So, there is a sequence of identically distributed independent random variables taking values on the integers, and they have a positive expectation. The problem is to prove that with probability 1 the sum from 1 to k of the random variables is 0 for finitely many k.

I was thinking to use the first Borel Cantelli Lemma, but I can't seem to show that the sum of the probabilities is finite.

Thanks in advance!

  • $\begingroup$ Actually this is the opposite, you need to check that $\sum P(X_n\ne0)$ diverges. $\endgroup$ – Did May 15 '16 at 20:48
  • $\begingroup$ You might have better luck with the Law of Large Numbers: If $S_k$ is the sum of the first $k$ of your random variables, and $\mu$ is their common mean, then $S_k/k\to\mu>0$ a.s. $\endgroup$ – John Dawkins May 15 '16 at 21:07
  • $\begingroup$ @Did that would be if I wanted to use the second Borel Cantelli Lemma, but that would require the events to be independent, which they are not $\endgroup$ – horseshoe nailpolish May 15 '16 at 22:04
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    $\begingroup$ Are they not? Quote: "a sequence of identically distributed independent random variables" $\endgroup$ – Did May 16 '16 at 7:51
  • $\begingroup$ @Did i think the proof should go in the way that assume $A_k=\{x:\sum_{i=1}^k X_i(x)=0\}$ and show that $\sum_{i=1}^{\infty}\mathbb P(A_i)<\infty$ then Borel Cantelli first lemma $\mathbb P(A_i i.o.)=0$ and hence $\mathbb P(A_k)=0$ for finitely many $k$ with probability $1$. $\endgroup$ – bunny Oct 18 '17 at 6:44

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