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Miller, SJ and Nigrini, MJ (2008)

The Modulo 1 Central Limit Theorem and Benford's Law for Products

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 L1([0, 1]), and discuss generalizations to discrete random variables. A consequence is that if X1, . . . , XM are independent continuous random variables with densities f1, . . . , fM, for any base B as M→∞ for many choices of the densities the distribution of the digits of X1 ∑ ∑ ∑XM converges to Benfordís 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 {fi} with the following properties: (1) for each i, if every Xk is independently chosen with density fi then the sum converges to the uniform distribution; (2) if the Xkís are independent but non-identical, with Xk having distribution fk, 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