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?

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

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