Convergence to the uniform distribution of vectors of partial sums modulo one with a common factor

TL;DR

This paper proves that vectors of partial sums of i.i.d. random variables, scaled by a common factor, converge in distribution to a uniform distribution on the unit cube, generalizing Benford's law and aiding in stratified resampling analysis.

Contribution

It establishes joint convergence of scaled partial sums modulo one to a uniform distribution, extending previous results and addressing coupling via a common factor.

Findings

Vectors of partial sums converge to uniform distribution on [0,1]^q

Results generalize Benford's law to data-adapted mantissas

Supports a central limit theorem for stratified resampling

Abstract

In this work, we prove the joint convergence in distribution of $q$ variables modulo one obtained as partial sums of a sequence of i.i.d. square integrable random variables multiplied by a common factor given by some function of an empirical mean of the same sequence. The limit is uniformy distributed over $[0,1]^q$. To deal with the coupling introduced by the common factor, we assume that the joint distribution of the random variables has a non zero component absolutely continuous with respect to the Lebesgue measure, so that the convergence in the central limit theorem for this sequence holds in total variation distance. While our result provides a generalization of Benford's law to a data adapted mantissa, our main motivation is the derivation of a central limit theorem for the stratified resampling mechanism, which is performed in the companion paper \cite{echant}.