# Problem that may use Borel Cantelli Lemma

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.

• Actually this is the opposite, you need to check that $\sum P(X_n\ne0)$ diverges. – Did May 15 '16 at 20:48
• 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. – John Dawkins May 15 '16 at 21:07
• Are they not? Quote: "a sequence of identically distributed independent random variables" – Did May 16 '16 at 7:51
• @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 $1$. – bunny Oct 18 '17 at 6:44