How to prove $\sum_{i\geq 1}N_i<\infty$ almost surely? Given $N_i$ are poisson rv with intensity $np_i$ where $\sum_{i\geq 1}p_i=1$

Question

How to prove $\sum_{i\geq 1}N_i<\infty$ almost surely? Given $N_i$ are poisson rv with intensity $np_i$ where $\sum_{i\geq 1}p_i=1$ and $n$ is given.

I have the above question when I am reading the paper Unbiased Estimators and Multilevel Monte Carlo (Example 1 after Theorem 4). It said that we can use Borel-Cantelli lemma to prove. I do not see how it goes..

Answer 1

This is just Borel Cantelli: Note that $E[\sum_i N_i] = \sum_i E[N_i] = \sum_i n p_i = n$. Hence $\sum_i N_i < \infty$ ae.

Note that the only facts used here are non negativity (to use Tonelli) and summability of the expectations (so the other integral is finite). (And measurability, of course!)