Equidistribution, Uniform distribution: a probabilist's perspective

TL;DR

This paper introduces probabilists to the theory of equidistribution, providing new examples, criteria, and perspectives that connect number theory with probability, especially in the context of uniform distribution and MCMC simulations.

Contribution

It offers a probabilistic perspective on equidistribution, introduces new examples of uniformly distributed sequences, and derives a Weyl-like criterion relevant for MCMC methods.

Findings

New examples of completely uniformly distributed sequences

A Weyl-like criterion for weakly completely equidistributed sequences

Connections between equidistribution and strong law of large numbers

Abstract

The theory of equidistribution is about hundred years old, and has been developed primarily by number theorists and theoretical computer scientists. A motivated uninitiated peer could encounter difficulties perusing the literature, due to various synonyms and polysemes used by different schools. One purpose of this note is to provide a short introduction for probabilists. We proceed by recalling a perspective originating in a work of the second author from 2002. Using it, various new examples of completely uniformly distributed (mod 1) sequences, in the "metric" (meaning almost sure stochastic) sense, can be easily exhibited. In particular, we point out natural generalizations of the original $p$-multiply equidistributed sequence $k^p\, t$ mod 1, $k\geq 1$ (where $p\in \mathbb{N}$ and $t\in[0,1]$), due to Hermann Weyl in 1916. In passing, we also derive a Weyl-like criterion for weakly completely equidistributed (also known as WCUD) sequences, of substantial recent interest in MCMC simulations. The translation from number theory to probability language brings into focus a version of the strong law of large numbers for weakly correlated complex-valued random variables, the study of which was initiated by Weyl in the aforementioned manuscript, followed up by Davenport, Erd\"{o}s and LeVeque in 1963, and greatly extended by Russell Lyons in 1988. In this context, an application to $\infty$-distributed Koksma's numbers $t^k$ mod 1, $k\geq 1$ (where $t\in[1,a]$ for some $a>1$), and an important generalization by Niederreiter and Tichy from 1985 are discussed. The paper contains negligible amount of new mathematics in the strict sense, but its perspective and open questions included in the end could be of considerable interest to probabilists and statisticians, as well as certain computer scientists and number theorists.