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Suppose $X_1, X_2, ...$ are independent random variables with $P(X_n=\sqrt n)=1/\sqrt n $ and $P(X_n=0)=1-1/\sqrt n$. Let $S_n=X_1+X_2+\cdots+X_n$ for all $n$. Show that $S_n/n \rightarrow 1 \space a.s.$
I try to apply the Borel-Cantelli lemma, but its not working. I also try to truncate the random variable by define a new random variable $Y_n=X_n\Large 1\normalsize\{ X_n<1\}$. But its still not working.
Can anybody help?

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    Where did you get this question? Was it in some book for example?2012-12-08

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I think it's easier to use the Strong Law of Large Numbers. The following theorem is useful:

Theorem (Kolmogorov's strong law) Let $X_j \in L^2$ independent random variables such that $\sum_{j \geq 1} \frac{\text{var}(X_j)}{j^2}<\infty$. Then $(X_j)$ fulfills the Strong Law of Large Numbers, i.e.

$\frac{1}{n} \sum_{j=1}^n (X_j-\mathbb{E}X_j)=0 \quad \text{a.s.}$

(See Sen & Singer (1993, Theorem 2.3.10)).

In this case we have $\text{var}(X_j) = \sqrt{j}-1 \leq \sqrt{j}$, hence $\sum_{j \geq 1} \frac{\text{var} X_j}{j^2}<\infty$. Thus (by Kolmogorov's strong law)

$ \frac{S_n}{n}- \underbrace{\frac{\mathbb{E}S_n}{n}}_{1} \to 0 \quad \text{a.s.}$

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    You could post it as an answer (to your own question) ... it would be interesting to see your way of solving it.2012-12-09