Convergence of a dominated series

Question

Let $X_1, X_2, \ldots$ and $Y_1, Y_2, \ldots$ be symmetric random variables with $|X_n| \leq |Y_n|$ such that the pairs $(X_n, Y_n)$ are independent. Suppose that $\sum_{n} Y_n$ converges almost surely. How do I show that $\sum_n X_n$ converges almost surely?

Answer 1

This method might be overkill, as it appeals to a high-power theorem.

Let $a\wedge b$ denote $\max(a,b)$. From the Kolmogorov Three series theorem, $\sum Y_n$ converges if and only if there exists a constant $A$ where $$ \sum P(|Y_n|>A)<\infty \qquad \sum E(Y_n1_{|Y_n\le A|})\,\,\text{ converges, and}\qquad\sum var(Y_n1_{|Y_n|\le A})<\infty $$ You can then use the fact that these conditions hold for $Y_n$ to prove that they hold for $X_n$, and deduce that $\sum X_n$ converges a.s.