# Convergence of series of symmetric random variables

Suppose $$X_i$$ are iid symmetric (about $$0$$) random variables. Then show that $$\sum_n\dfrac{X_n}{n}$$ converges almost surely if and only if $$E|X_1|<\infty$$.

I was thinking of applying the Kolmogorov 3-series theorem. It states that if $$Y_n$$ are independent then $$\sum_n Y_n$$ converges a.s. if and only if there exists a constant $$c>0$$ such that $$\sum_n P(|Y_n|>c)<\infty,\sum_n Var(Y_n 1(|Y_n|\leq c))<\infty$$ and $$\sum_n E(Y_n 1(|Y_n|\leq c))$$ converges.

Suppose $$\sum_n X_n/n$$ converges. Then there exists $$c>0$$ with $$\sum_n P(|X_n|>nc)<\infty$$ and since $$X_i$$ are iid, $$\sum_n P(|X_1|>nc)<\infty$$ implying $$E|X_1|/c<\infty$$ implying $$E|X_1|<\infty$$.

I am stuck in proving the converse. Suppose $$E|X_1|<\infty$$. This implies for any $$c>0$$, $$\sum_n P(|X_n|>nc)<\infty$$ and since $$X_1$$ is symmetric, $$E(X_11(|X_1|\leq nc))=0$$ for each $$n$$ and thus $$\sum_n E(X_n1(|X_n|\leq nc))$$ converges.

But I cannot show $$\sum_n Var(X_11(|X_1|\leq nc))<\infty$$.

• Law of large numbers is fair game ? Apr 29 '18 at 13:34
• No, you probably misinterpret it as $\sum_{1\leq k\leq n}X_k/n$ whereas I mean the infinite random sum $\sum_n X_n/n$. Apr 29 '18 at 14:06
• Ah yes, sorry then. Apr 29 '18 at 14:16

We have to prove the convergence of the series $$\sum_n\color{red}{\frac 1{n^2}}\mathbb E\left[X_1^2\mathbf 1\left\{\left\lvert X_1\right\rvert \leqslant n\right\}\right].$$ Letting $a_k:=\mathbb E\left[X_1^2\mathbf 1\left\{k-1\lt \left\lvert X_1\right\rvert \leqslant k\right\}\right]$, we have to prove the convergence of $$\sum_{n=1}^{\infty}\frac 1{n^2} \sum_{k=1}^na_k,$$ which can be done by summing first over $n\geqslant k$ and then over $k$.