Let $X_n$ be random variables such that for some $a_n\in \mathbb{R}$:
\begin{align}
\sum\limits_{n\ge1}\mathbb{P}(X_n\ne a_n)<\infty \quad \text{and} \quad \sum\limits_{n\ge1}a_n \ \ \text{converges}
\end{align}
Show that $\sum\limits_{n\ge1}X_n$ converges a.s.
I feel like I don't know a whole lot about how to show that a series of random variables converges almost surely, so any help on this one or even general techniques to show almost sure convergence of a series would be greatly appreciated.
Best Answer
Let $E_n=\{X_n\ne a_n\}$ and $E=\cap_{n=1}^\infty \cup_{k=n}^\infty E_k$. Note that if $\omega\notin E$ then we have $X_n(\omega)\ne a_n$ for only finitely many values of $n$, so the sequence $X_n(\omega)$ eventually just becomes the sequence $a_n$, and hence $\sum_{n=1}^\infty X_n(\omega)$ converges. So now we just have to prove that the probability of the event $\{\omega\notin E\}$ is $1$, or equivalently that $\mathbb{P}(E)=0$. But this just follows immediately from the first Borel-Cantelli lemma.