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Consider random variable $X$ formed by the following infinite series: $X = \pm 1 \pm \frac{1}{2} \pm \frac{1}{3} \pm ... \frac{1}{n} ...$, where $+$ or $-$ sign for every summand is chosen independently w.p. $1/2$. What is the distribution of $X$? If it is not some well-known distribution, does it have any interesting properties?

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There might be no better answer than to direct you to this paper (which happens to be written by a contributor to this site): Random harmonic series, American Mathematical Monthly 110, 407-416, 2003.

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    $\begingroup$ very nice paper! $\endgroup$ Apr 22 '12 at 12:13

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