Let $(X_n)_{n\ge 1}$ be a sequence of real valued random variables defined on some probability space $(\Omega, \mathcal{A},P)$. Assume that there exists a sequence of real numbers $(a_n)_{n\ge 0}$ such that both series
$$\sum_{n\ge 0}a_n \quad and \quad\sum_{n\ge 0}P(X_n\neq a_n)$$
converge. Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely.
I don't really know how to solve this problem. The first thing that I don't understand is the sentence
Prove that the series $\sum\limits_{n=0}^{\infty}X_n$ converges almost surely.
What is the series intended to converge to? I've thought it is intended to converge almost surely to $0$, so that the statement to prove would be
$$P\left(\lim_{n\to \infty}\sum_{n=0}^{\infty}X_n =0 \right)=1$$
But then i can't really move on. I was thinking to use Strong Law of Large Numbers, but to use it the sequence of random variables must be i.i.d. and in the Problem they are not. Do you have any hint?
Best Answer
By "the series $\sum\limits_n X_n$ converge almost surely", it is meant that there exists $\Omega'$ of probability $1$ for which if $\omega\in\Omega'$, then the series of real numbers $\sum\limits_{n=1}^{+\infty}X_n(\omega)$ is convergent.
Define the event $E_n:=\{X_n\neq a_n\}$. We get by the assumption that $\mathbb P\left(\limsup\limits_nE_n\right)=0$. Can you show that if $\omega$ belongs to the complement of $\limsup\limits_nE_n$, then the series $\sum\limits_{n=1}^{+\infty}X_n(\omega)$ is convergent?