H.F.D. (H-function Distribution) and Benford's Law. I

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 $\alpha$ to 0. 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.