International Journal of Algebra 2(3), pp. 119 - 130.
ISSN/ISBN: Not available at this time. DOI: Not available at this time.
Abstract: Using elementary results from Fourier analysis, we provide an alternate proof of a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distribution in L^{1}([0, 1]), and discuss generalizations to discrete random variables. A consequence is that if X_{1}, . . . , X_{M} are independent continuous random variables with densities f_{1}, . . . , f_{M}, for any base B as M→∞ for many choices of the densities the distribution of the digits of X_{1} · · ·X_{M} converges to Benford’s law base B. The rate of convergence can be quantified in terms of the Fourier coefficients of the densities, and provides an explanation for the prevalence of Benford behavior in many diverse systems. To highlight the difference in behavior between identically and non-identically distributed random variables, we construct a sequence of densities {f_{i}} with the following properties: (1) for each i, if every X_{k} is independently chosen with density f_{i} then the sum converges to the uniform distribution; (2) if the X_{k}’s are independent but non-identical, with X_{k} having distribution f_{k}, then the sum does not converge to the uniform distribution.
Bibtex:
@article{,
title={The modulo 1 central limit theorem and Benford's law for products},
author={Miller, Steven J and Nigrini, Mark J},
journal={International Journal of Algebra},
volume={2},
number={3},
pages={119-130},
year={2008}
}
Reference Type: Journal Article
Subject Area(s): Probability Theory