TITLE: A Qual Problem:Convergence of sum of Poisson random variable over sum of their mean
QUESTION [5 upvotes]: Assume that $X_i$'s are independent  Poisson$(\lambda_i)$, and $\sum_i^\infty\lambda_i=\infty$. Show that $\frac{\sum_i^n X_i}{\sum_i^n\lambda_i}\rightarrow1$ almost surely.
I also tried to show weaker version: convergence in probability, yet I could not succeed. We know that $\sum_i^\infty\lambda_i=\infty$ implies $\sum_i^\infty X_i=\infty$ almost surely but i could not connect their rate of convergence to solve the problem.
For any help/hint/solution, thanks in advance.

REPLY [2 votes]: What a coincidence... This problem was on my probability qual just a few days ago.
I think it's pretty straightforward when you use the version of Strong Law of Large Numbers that uses Kronecker's Lemma.
Or just embedding the sum in a Poisson process...