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Suppose $X_{n}$ is any sequence of (real-valued) random variables defined on a common probability space $(\Omega,\mathcal{F},\mathbb{P})$. Show that there exists a sequence $c_{n}\rightarrow \infty$ such that $\frac{X_{n}}{c_{n}}\rightarrow 0$ almost surely. The key is to use the Borel Cantelli lemma, but I am not able to formulate pairwise independent events $A_{n}$ such that $\sum\limits_ {n=1}^{\infty}\mathbb{P}(A_{n})=\infty$ implies $\mathbb{P}(A_{n}~\text{i.o.})=1$, where i.o. stands for infinitely often.

Just a few small hints would be greatly appreciated.

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Use (or reprove) the extended form of Borel Cantelli, which says that if there exists a decreasing sequence $\epsilon_i\rightarrow 0$ such that $\sum_i P(|X_i|>\epsilon_i)<\infty$, then $X_i$ converges to 0 almost surely. You can pick any such sequence $\epsilon_i$, such as $\epsilon_n=1/n$, and then pick a corresponding $c_n$ sequence, that decreases $P(|X_n|>c_n/n)$ sufficiently fast.

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  • $\begingroup$ Thank you so much!! Will try the problem with this. $\endgroup$ – PN Karthik Sep 3 '16 at 18:32

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