1
$\begingroup$

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?

$\endgroup$
1
$\begingroup$

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 $i$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.