Suppose $X_n$ are independent random variables with $P(X_n=-n^2)=\dfrac{1}{n^2}$, $P(X_n=-n^3)=\dfrac{1}{n^3}$ and $P(X_n=2)=1-\dfrac{1}{n^2}-\dfrac{1}{n^3}$ for all $n\geq2$. Let $S_n=\sum_{i=1}^nX_i$. Show that $S_n\to\infty$ almost surely.

I cannot find any result relating sums of random variables to variances or means. Help is appreciated.


Hint: use the Borel-Cantelli lemma to show that $$P(X_n \ne 2 \text{ i.o.}) = 0.$$ (Independence is not needed,)

  • 1
    $\begingroup$ @LandonCarter: This shows that almost surely, $X_n = 2$ for all but finitely many $n$. For such a sequence, what can you say about $\lim_{n \to \infty} \sum_{i=1}^n X_n$? $\endgroup$ – Nate Eldredge Aug 22 '15 at 5:50

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