Geodesic
H.F.D.\ ($H$-function distribution) and the Benford law.~I
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 2, pp. 366-371

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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, Benford law
Mots-clés : gamma-distributions, Poisson summation formula. 
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     author = {A. A. Kulikova and Yu. V. Prokhorov and V. I. Khokhlov},
     title = {H.F.D.\ ($H$-function distribution) and the {Benford} {law.~I}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     year = {2005},
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A. A. Kulikova; Yu. V. Prokhorov; V. I. Khokhlov. H.F.D.\ ($H$-function distribution) and the Benford law.~I. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 2, pp. 366-371. http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a9/