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Wojcik, MR (2013)

Notes on scale-invariance and base-invariance for Benford's Law

arXiv:1307.3620 [math.PR].

ISSN/ISBN: Not available at this time. DOI: Not available at this time.



Abstract: It is known that if X is uniformly distributed modulo 1 and Y is an arbitrary random variable independent of X then Y+X is also uniformly distributed modulo 1. We prove a converse for any continuous random variable Y (or a reasonable approximation to a continuous random variable) so that if X and Y+X are equally distributed modulo 1 and Y is independent of X then X is uniformly distributed modulo 1 (or approximates the uniform distribution equally reasonably). This translates into a characterization of Benford's law through a generalization of scale-invariance: from multiplication by a constant to multiplication by an independent random variable. We also show a base-invariance characterization: if a positive continuous random variable has the same significand distribution for two bases then it is Benford for both bases. The set of bases for which a random variable is Benford is characterized through characteristic functions.


Bibtex:
@unpublished{, title={Notes on scale-invariance and base-invariance for Benford's Law}, author={Wojcik, Michal Ryszard}, eprint = {arXiv:1307.3620 [math.PR]}, year={2013} note = {last accessed 6/20/2014}, url = {http://arxiv.org/abs/1307.3620}, }


Reference Type: E-Print

Subject Area(s): Analysis, Probability Theory