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I'm working on the problem below. I've proved one side of it, but I need help on the other side.

Consider $X_1, X_2, \ldots$ as independent random variable where: $\Pr(X_n = k) = (1-p_n)p_n^k$ for $k = 0, 1, 2, \ldots$ ($p_n > 0$)

The goal is to show:

$$X_n \rightarrow 0 \text{ almost surely IF and Only IF }\sum_{n} P_n < \infty$$

1) part 1: $\sum_{n} P_n < \infty \rightarrow X_n \rightarrow 0 $

To prove this I can simply show that $\sum_{K = 0}^{\infty}P(X_n) = K = 1 < \infty$ then by using Borel Cantelli we know this implies that $P(|X_n| > \epsilon, \text{infinitely often }) = 0 \rightarrow X_n \text{ converges to 0 almost surely}$

Could you guide me whether my solution to first part is correct. Also,could you guide me on the second part?

I appreciate your help.

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up vote 1 down vote accepted

Assume that $\sum_n p_n<\infty$. Let $E_n:=\{\omega,X_n(\omega)\geqslant 1\}$. Then $P(E_n)=p_n$ so By Borel-Cantelli lemma, $P(\limsup_{n\to +\infty}E_n)=0$ which gives almost sure convergence.

If $\sum_n p_n=+\infty$, as the events $E_n$ are independent, by the converse result of Borel-Cantelli lemma, $P(\limsup_{n\to +\infty}E_n)=1$ which proves we don't have almost sure convergence.

(here, we used the fact that the random variables have integer values)

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