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Suppose we have an infinite sequence $X_i^{(n)}$ of random variables such that for each $i$, $X_i^{n} \overset{a.s.}{\to} 0$ as $n \to \infty$. With this in hand, which additional conditions need to hold (if any) to ensure that $$\sum_{i=1}^{\infty}X_i^{n} \overset{a.s.}{\to} 0$$ as $n \to \infty$?

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  • $\begingroup$ It looks like you need some condition for large values of $I$ knowing that $\sum_{i=1}^I X_i^n\to 0$, a.e. for all $I$. $\endgroup$ Jun 13, 2019 at 21:40
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    $\begingroup$ You definitely need much stronger conditions than currently stated. If each sequence starts with $X_1^n = 1$ then each individual sequence can still converge to $0$ almost surely, while the sum can be greater than 1 infinitely often $\endgroup$
    – Xiaomi
    Jun 14, 2019 at 4:57
  • $\begingroup$ @Xiaomi: Thanks, that's very insightful! Should have thought about that myself $\endgroup$
    – Jeremias K
    Jun 14, 2019 at 14:58

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