# Independent random variables convergence almost surely

I'm preparing for a final exam and I'm working through a practice exam. I'm somewhat stuck on a problem.

The random variables $\{X_i\}$ are all independent and all satisfy $E[X_i^4]\leq 1.0$, but may have different distributions. Let $S_n\equiv\sum_{i=1}^n X_i$ be their partial sum. Does $S_n/n$ converge almost surely?

Are there extra conditions that are needed to say that the quantity $S_n/n$ converges a.s.? We know $X_i\in L_2$ since $L_4\subset L_2$. Is it enough to show that the sum of the variances (in the form of $\sum EX_i^2 < \infty$) is finite?

## 1 Answer

Does $S_n/n$ converge almost surely?

Not necessarily, consider deterministic random variables such that $X_i=1$ if $4^k\leqslant i\lt2\cdot4^k$ and $X_i=-1$ if $2\cdot4^k\leqslant i\lt4^{k+1}$, for every $k\geqslant0$. The first values of $(X_i)_{i\geqslant1}$ are $+1$ once then $-1$ twice then $+1$ four times then $-1$ eight times, and so on, hence $S_n/n$ oscillates indefinitely between $\frac13$ and $-\frac13$.

Are there extra conditions that are needed to say that the quantity $S_n/n$ converges a.s.?

Yes, see above.

Is it enough to show that the sum of the variances (in the form of $\sum EX_i^2 < \infty$) is finite?

No, consider once again deterministic random variables with $X_i=\frac1i$ for every $i$.