Convergence almost surely - Example

Question

Give an example of string {Xn} from n = 1 to infinity independent random variables with zero expected value such that arithmetic average ( sum Xn divided by n) → $ - \infty$ almost surely. It is possible? I use strong law of large numbers and something is wrong :/

Answer 1

The key point is that the law of large numbers requires the random variables to be identically distributed. That assumption is lacking here.

For example, try $X_n = 2^{2n}$ with probability $2^{-n}$, $-2^n/(1-2^{-n})$ with probability $1 - 2^{-n}$.

Since $\sum_{n=1}^\infty 2^{-n}$ converges, with probability $1$ there are only finitely many positive $X_n$; when this happens, there is some finite $c$ such that $\sum_{n=1}^N X_n \le c - 2^N$, and so the average goes to $-\infty$.