The Benford phenomenon for random variables. Discussion of Feller's way

TL;DR

This paper surveys the Pinkham-Feller approach to the Benford phenomenon, introduces new generalizations, and discusses the impact of high-frequency oscillations on the distribution of significant digits.

Contribution

It extends previous results by removing density assumptions and explores the effects of oscillations, providing new insights into the conditions for Benford's law to hold.

Findings

Theorem 2 generalizes previous results without requiring density.

Section 10 discusses the influence of high-frequency oscillations.

The survey clarifies the Pinkham-Feller approach and offers new perspectives.

Abstract

This is a detailed survey which mainly presents the Pinkham-Feller way. I added some new points to the first version [V2] and I suppressed "Examples" devoted to Gamma, Fr\'echet and Weibull laws. Theorem 2 is a bit more general (no assumption of density: this answers a question of T. Hill). Section 10 is new and devoted to an argument (Poincar\'e, Fewster) about the effect of high frequencies oscillations. Maybe many works, many efforts, have been devoted to the study of a sufficient condition of poor value: see Sections 7 and 11. The final Section gives some suggestions.