I'm trying to solve the question below and after an hour no success yet! Here is the question:
$X_1, X_2, ...$ are independent poisson random variables with $EX_j = \lambda_j$. Assume that $ 0 < \lambda_j < 1$. Define $S_n = \sum_{i = 1}^{n} X_i$. We need to show that:
$ \text{if } \sum_{j = 1}^{\infty} \lambda_j = \infty \text{ then} \frac{S_n}{ES_n} \rightarrow 1 \text{ almost surely}$.
I appreciate if you could give me some hints on how I should approach this question.
There is a hint on this question that Borel-Cantelli lemma should be used. As you know Borel-Cantelli (lemma 2) says:
Let $A_1, A_2, ...$ be events in the probability space. Then:
if $A_1, A_2, ...$ are independent and $\sum_{k = 1}^{\infty} p(A_k) = \infty$ then $\rightarrow p(A_k, \text{ infinitely often}) = 1$
Thank you for your help.