Geodesic
On one identity for distribution of sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 2, pp. 396-397 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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V. M. Kruglov. On one identity for distribution of sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 58 (2013) no. 2, pp. 396-397. http://geodesic.mathdoc.fr/item/TVP_2013_58_2_a10/

[1] Spitzer F., “The Wiener–Hopf equation whose kernel is a probability density”, Duke Math. J., 24:3 (1957), 327–343 | DOI | MR | Zbl

[2] Barsegyan A. G., “O raspredelenii summ odinakovo raspredelennykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 57:1 (2012), 155–158 | DOI | Zbl

[3] Petrov V. V., Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Nauka, M., 1987, 317 pp.