[Math] Almost sure convergence using Borel-Cantelli lemma

Question

Let $(X_n)$ be a sequence of random variables. I want to show that if $E[X_n] \rightarrow C$ and $Var(X_n) \leq \frac{C}{n^2}$, where $C$ is some constant, then $X_n$ converge almost surely to $C$.

I use Borel-Cantelli Lemma, i.e. $$P\left[|X_n – E[X_n]| > \frac{1}{\sqrt[4]{n}}\right] \leq \sqrt{n} Var(X_n) \leq \frac{C}{n^{\frac{3}{2}}}$$ Hence by Borel-Cantelli lemma $$P\left[|X_n-E[X_n]| > \frac{1}{\sqrt[4]{n}} i.o.\right] = 0$$ On the other hand, $|E[X_n]-C| < \frac{1}{\sqrt[4]{n}}$ for some large $n$. And my claim will follow?

Answer 1

Your proof is correct. More generally:

If $E(X_n)\to C$ and the series $\sum\limits_n\mathrm{var}(X_n)$ converges, then $X_n\to C$ almost surely.

Hint: There exists some positive sequence $(\alpha_n)$ such that $\alpha_n\to0$ and $\sum\limits_n\frac1{\alpha_n^2}\mathrm{var}(X_n)$ converges. Consider the events $A_n=[|X_n-E(X_n)|\geqslant\alpha_n]$.