1. ## converges almost surely

Let X1, X2,... be iid having expected value and a symmetric distribution
P(X1>x)=P(X1<-x) for all x. Show that SUM_n (Xn/n) converges almost surely.

2. Originally Posted by economics
Let X1, X2,... be iid having expected value and a symmetric distribution
P(X1>x)=P(X1<-x) for all x. Show that SUM_n (Xn/n) converges almost surely.
If $X_1,X_2,\ldots$ are iid and have finite expectation then

$\sum_n\frac{X_n}{n}$ converges a.s. by the SLLN. This implies your problem