Almost sure convergence of sum of random variables

Question

Suppose $X_i$ are mutually independent random variables such that $P(X_n=n^2-1)=1-P(X_n=-1)=n^{-2}$ for $n=1,2,3,...$. Show that $E(X_n)=0$ for all n, while $n^{-1}\sum\limits_{i=1}^n X_i\rightarrow -1$ a.s. for $n\rightarrow \infty$.

i am done with proving $E(X_n)=0$ but do not have any idea about how to approach it. Any type of help will be appreciated. Thanks in advance.

Answer 1

For the second part of the result: use the Borel-Cantelli Lemmas to show that with probability 1, $X_n = -1$ for all but finitely many $n$. Then, use that fact to show that $\overline {X_n}$ converges to $-1$ almost surely.

Answer 2

For showing $E(X_n)=0$ check that $X_n=(-1)\chi_{X_n=-1}+(n^2-1)\chi_{X_n=n^2-1}$ for all $n$ and hence $E(X_n)=(-1)P(X_n=-1)+(n^2-1)P(X_n=n^2-1)=(-1)(1-\frac{1}{n^2})+(n^2-1)\frac{1}{n^2}=0$