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Short Communications

On one identity for distribution of sums of independent random variables

V. M. Kruglov

M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: In a 1957 paper F. Spitzer published, among other things, one estimate of the distribution function of sums of independent symmetric random variables with a common absolutely continuous distribution. This estimation was constructed with the help of one identity for the distribution of a sum of the above-mentioned random variables, the proof of which was not given. In this paper we prove the Spitzer identity for any independent random variables, and by using it we construct an estimate of the distribution of a sum of independent identically distributed random variables. The Spitzer estimate can be derived as a particular case of the proposed estimation.

Keywords: independent random variables; characteristic function; inverse formula.

MSC: 60

Received: 05.07.2012

DOI: 10.4213/tvp4513

 English version: Theory of Probability and its Applications, 2014, 58:2, 329–331

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