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 Teor. Veroyatnost. i Primenen., 2005, Volume 50, Issue 2, Pages 366–371 (Mi tvp113)

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H.F.D. ($H$-function distribution) and the Benford law. I

A. A. Kulikovaa, Yu. V. Prokhorovb, V. I. Khokhlovb

a M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics
b Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: This paper notes a connection among a wide class of the so-called $HF$-random variables, approximately uniform distributions, and Benford's law. This connection is considered in detail with the help of examples of random variables having gamma-distribution. Let $Y$ be a random variable having gamma-distribution with parameter $\alpha$. It is proved that the distribution of a fractional part of the logarithm of $Y$ with respect to any base larger than 1 converges to the uniform distribution on the interval $[0,1]$ for $\alpha\to0$. This implies that the probability distribution of the first significant digit of $Y$ for small $\alpha$ can be approximately described by Benford's law. The order of the approximation is illustrated by tables.

Keywords: $H$-function distribution, gamma-distributions, Benford law, Poisson summation formula.

DOI: https://doi.org/10.4213/tvp113

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English version:
Theory of Probability and its Applications, 2006, 50:2, 311–315

Bibliographic databases:

Citation: A. A. Kulikova, Yu. V. Prokhorov, V. I. Khokhlov, “H.F.D. ($H$-function distribution) and the Benford law. I”, Teor. Veroyatnost. i Primenen., 50:2 (2005), 366–371; Theory Probab. Appl., 50:2 (2006), 311–315

Citation in format AMSBIB
\Bibitem{KulProKho05} \by A.~A.~Kulikova, Yu.~V.~Prokhorov, V.~I.~Khokhlov \paper H.F.D.\ ($H$-function distribution) and the Benford law.~I \jour Teor. Veroyatnost. i Primenen. \yr 2005 \vol 50 \issue 2 \pages 366--371 \mathnet{http://mi.mathnet.ru/tvp113} \crossref{https://doi.org/10.4213/tvp113} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2221718} \zmath{https://zbmath.org/?q=an:1098.60021} \elib{http://elibrary.ru/item.asp?id=9153128} \transl \jour Theory Probab. Appl. \yr 2006 \vol 50 \issue 2 \pages 311--315 \crossref{https://doi.org/10.1137/S0040585X97981706} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000238760000011} 

• http://mi.mathnet.ru/eng/tvp113
• https://doi.org/10.4213/tvp113
• http://mi.mathnet.ru/eng/tvp/v50/i2/p366

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. Ya. Kuznetsova, “On mixing of unimodal distributions”, Theory Probab. Appl., 51:3 (2007), 535–536
2. O. V. Viskov, V. I. Khokhlov, “Four areas of Yu. V. Prokhorov's studies and their perspectives”, Theory Probab. Appl., 60:2 (2016), 336–342
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