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Let $\{X_i\}$ be a uniformly bounded sequence of independent random variables. Does $\sum_{i=1}^{\infty}X_i-E[X_i]$ converges or diverges, depending on whether $\sum_{i=1}^{\infty}\sigma_i^{2}$ converges or diverges?

I have looked everywhere for a proof with little success. Can someone provide a reference or a proof? Thank you.

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  • $\begingroup$ There's a whole book "Almost sure convergence", by Stout, where this must (almost) surely be one of the first results. $\endgroup$
    – Christian Remling
    Commented Nov 14, 2019 at 20:42
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    $\begingroup$ Kolmogorov's three-series theorem is your friend, if you ask for almost sure convergence. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Nov 14, 2019 at 20:43

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