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Suppose $X$ is a random variable and $\{a_n\}$ is a real sequence converging to 0 as $n\to \infty$. It's clear to me that if $\sum_{i=0}^\infty a_i^2<\infty$ then the sequence formed by $a_nX$ converges almost surely to zero. What if this last condition is not satisfied? I can't prove either convergence almost surely neither find a counterexample.

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  • $\begingroup$ What do you mean by "the sequence formed by $a_n X$"? Is every $a_n$ multiplied by a different trial of $X$? $\endgroup$ – Holding Arthur Apr 15 at 13:05
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Use $$a_n=\frac1{\sqrt{n^2+1}}$$ $$X\sim \text{Geo}(0.5)$$ as a counterexample.

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