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Let $X_n$ be a sequence of independent random variables such that $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Consider the sum $S_n=\Sigma_{k=1}^nX_k.$

How can we show that for any $\epsilon>0$ $$\frac{S_n}{n^{1/2+\epsilon}}$$ converges to $0$ almost surely?

Moreover, how can we prove that $\frac{S_n}{n^{1/2}}$ does not converge to $0$ in probability?

I have done some work but I dont think its correct. Can someone please help me?

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  • $\begingroup$ Try computing the variance. Overall, you could closely follow a proof you know of the Law of Large numbers. $\endgroup$ – Michael Apr 7 '15 at 4:32
  • $\begingroup$ @Michael this is not really helpful. But I have tried using Chebyshev's inequality to show that the series $\Sigma \mathbb{P}(|S_n/n^{1/2+\epsilon}|<\epsilon)$ goes to 0 etc... But no use $\endgroup$ – L.G Apr 7 '15 at 4:35
  • $\begingroup$ Why is it no use? And perhaps first show a sparsely sampled sequence goes to zero (which is a standard proof for LLN). Like take $n=(1+\epsilon)^k$. $\endgroup$ – Michael Apr 7 '15 at 4:36
  • $\begingroup$ What if it is sparsely going to 0? Does it mean the series itself goes to 0? $\endgroup$ – L.G Apr 7 '15 at 4:39
  • $\begingroup$ If you do not know that proof, then is this a problem set exercise that asks you to use some particular method you know? How did you prove LLN? Fourth moment? $\endgroup$ – Michael Apr 7 '15 at 4:43

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