Products of random variables and the first digit phenomenon

TL;DR

This paper establishes conditions under which the mantissa of products of dependent, non-stationary random variables follows or converges to Benford's law, extending understanding of digit distribution in complex stochastic processes.

Contribution

It introduces new conditions for dependent, non-stationary variables to exhibit Benford's law in their product mantissas, broadening previous results.

Findings

Conditions for mantissa distribution to follow Benford's law

Almost sure convergence to Benford's law for product sequences

Extension to dependent and non-stationary random variables

Abstract

We provide conditions on dependent and on non-stationary random variables $X_n$ ensuring that the mantissa of the sequence of products $\left(\prod_{1}^{n}X_k\right)$ is almost surely distributed following the Benford's law or converges in distribution to the Benford's law.