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Let $(X_n)_{n\geq1}$ be a sequence of i.i.d. $\sim \text{Uni}([0,1])$ distributed random variables. I want to show that

$$\mathbb{P}\big(n^2 X_{n+1} < \sum\limits_{k=1}^n X_k ~\text{for infinitely many } n \in \mathbb{N} \big)=1 $$

This cries out for Borel Cantelli, but I don't know how to use this theorem when there is more than one random variable involved. By what means can I compare the two sides?

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Are you sure about the equation? It doesn't pass expectation test. –  karakfa Jul 17 '12 at 18:06
    
use the SLLN on the rhs –  mike Jul 17 '12 at 18:38

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This is an application of Lévy's conditional form of Borel-Cantelli lemma. The result is often stated as follows. Consider a sequence of events $(A_n)_n$ which is adapted to a given filtration $(\mathcal F_n)_n$. Then the random series $\sum\limits_n\mathbf 1_{A_n}$ converges/diverges almost surely if and only if the random series $\sum\limits_n\mathrm P(A_{n+1}\mid\mathcal F_{n})$ converges/diverges almost surely.

Here, consider $\mathcal F_n=\sigma(X_k;k\leqslant n)$ and $A_{n+1}=[n^2X_{n+1}\leqslant S_n]$ with $S_n=X_1+\cdots+X_n$. Then $\mathrm P(A_{n+1}\mid\mathcal F_{n})=\frac1{n^2}S_n$. By the strong law of large numbers, $\frac1nS_n\to\mathrm E(X_1)=\frac12$ hence, almost surely, $S_n\gt\frac14n$ for every $n$ large enough. This proves that $\sum\limits_n\frac1{n^2}S_n$ diverges almost surely. Hence, almost surely, infinitely many events $A_n$ occur, QED.

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