Related question not treated in Schmulands article: What is known about the distribution of Z = e_1 + 1/3 e_2 + 1/5 e_3 + 1/7 e_4 + .... with e_i i.i.d. +/-1? Remark: If X is the random harmonic series, then X = Y * Z, where Y is equally distributed in (-1,1). Any hints would be appreciated. Cheers, Hartmut asimovd@aol.com@mailman.xmission.com on 06.05.2003 11:06:04 Bitte antworten an math-fun@mailman.xmission.com Gesendet von: math-fun-bounces@mailman.xmission.com An: math-fun@mailman.xmission.com Kopie: Thema: Re: [math-fun] Probabilistic harmonic series The question was, What is the distribution of the sum of the random harmonic series ±1 ± 1/2 ± 1/3 ± ... ? By amazing synchronicity, an article on this very subject, "Random harmonic series" by Byron Schmuland, appears in the May, 2003 issue of the American Mathematical Monthly and is available from the author's website. (Thanks, Neil, for pointing this out.) As E. Clark mentioned, the article's available for downloading at <http://www.stat.ualberta.ca/people/schmu/publications.html>. I haven't studied every detail of this paper, but it's very interesting, with a number of graphs of the distribution function and related functions (like the distribution function if the same question is asked about placing random signs in front of the series of *squares* of reciprocal integers). This paper rewards whatever level of perusal you feel like giving it. It uses a clever series-rearrangement trick to nail down the distribution function in question. The author proves that this distribution function is infinitely differentiable, though he wrote me in private e-mail that for 15 years before solving the question, he believed it would not even be differentiable at all. He wrote that he did not know whether it is real analytic. In case anyone is interested, I'll mention that I've proved it's in fact real analytic. --Dan ------------------------------------------------------------------------------- ------------------------------------------------------------------------------- _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun -------------------------------------------------------------------------------