Convergence to the Uniform Distribution of Vectors of Partial Sums Modulo One with a Common Factor

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 uniformly distributed over \([0,1]^q\). To deal with the coupling introduced by the common factor, we assume that the absolutely continuous (with respect to the Lebesgue measure) part of the joint distribution of the random variables is nonzero, 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 (Flenghi and Jourdain, Central limit theorem for the stratified selection mechanism, 2023, http://arxiv.org/abs/2308.02186).