# Weak Law of Large Numbers for Arbitrary r.v.'s

I am recently stuck by a weak LLN for arbitrary r.v.'s:

For any sequence of r.v.'s $\{X_n\}$, if $E(X_n^2) \rightarrow 0$, then $$\frac{S_n-E(S_n)}{n}\rightarrow 0 \quad \text{in probability},$$ where $S_n=\sum_{j=1}^nX_j$.

Without loss of generality, we assume $E(X_j)=0$ for each $j>0$. It follows from Chebyshev's inequality that, to prove the result above, we need to prove $$E(S_n^2)=\sum_j E(X_j^2)+2\sum_{1\leq i<j\leq n}E(X_jX_k)=o(n^2).$$ By $E(X_n^2) \rightarrow 0$, it is clear that $E(X_n^2)$ is uniformly bounded. Thus the only trouble remained is $\sum_{1\leq i<j\leq n}E(X_jX_k)$.

How should I proceed?

• Completely wild guess: What happens if you use Cauchy-Schwarz? – Nate Eldredge Sep 5 '13 at 5:12
• What happens is that this proves the result since $a_n=o(1)$ implies $\left(\sum\limits_{k=1}^na_k\right)^2=o(n^2)$. – Did Sep 5 '13 at 14:34