On Benford's law to variable base
type: Item
instance of: scholarly article
title: On Benford's law to variable base
zbMATH Open document ID: 0908.60018
DOI: 10.1016/S0167-7152(97)00142-9
published in: Statistics \& Probability Letters
publication date: 1998-10-06T00:00:00Z
review text: Given two real numbers \(y\neq 0\) and \(b>1\), we can write \(| y|= mb^k\), where \(1\leq m<b\) and \(k\) is an integer. Here \(m=m(y,b)\) is called the mantissa of \(y\), while \(k=k (y,b)\) is called the exponent of \(y\) to the base \(b\). If \(y\) is a random variable, then the distribution function \(M(b,x): =P[m(y,b) <x]\) is called the mantissa distribution of \(y\) to the base \(b\). In a number of cases, \(m\) is logarithmically distributed: \(M(b,x)= \log_bx\) for \(1<x <b\). This phenomenon is called Benford's law to the base \(b\). From the main result of the present paper it follows that the approximation by Benford's law becomes worse as \(b\to\infty\), but it gets better as \(b\to 1+0\).
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 60E99
zbMATH DE Number: 1160171
zbMATH Keywords: first-digit problem
zbMATH Keywords: Benford's law
zbMATH Keywords: products of random variables
zbMATH Keywords: sums of random variables
zbMATH Keywords: discrepancy
author: Peter Schatte
reviewed by: Ferenc Móricz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1160171