A characterization of Benford’s law through generalized scale-invariance

• We characterize Benford’s law through generalized scale-invariance.
• Multiplication by a constant is replaced with multiplication by a random variable.
• Continuous and discrete random variables are considered.

If XX is uniformly distributed modulo 1 and YY is independent of XX then Y+XY+X is also uniformly distributed modulo 1. We prove a converse for any continuous random variable YY (or a reasonable approximation to a continuous random variable) so that if XX and Y+XY+X are equally distributed modulo 1 and YY is independent of XX then XX 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.