Open Access
2018 Equidistribution, uniform distribution: a probabilist’s perspective
Vlada Limic, Nedžad Limić
Probab. Surveys 15: 131-155 (2018). DOI: 10.1214/17-PS295


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 $\mathsf{mod}~1$ 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 $k^{p}\,t\ \mathsf{mod}~1$, $k\geq1$ (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, Erdős and LeVeque in 1963, and greatly extended by Russell Lyons in 1988. In this context, an application to $\infty$-distributed Koksma’s numbers $t^{k}\ \mathsf{mod}~1$, $k\geq1$ (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.


Download Citation

Vlada Limic. Nedžad Limić. "Equidistribution, uniform distribution: a probabilist’s perspective." Probab. Surveys 15 131 - 155, 2018.


Received: 1 November 2017; Published: 2018
First available in Project Euclid: 24 April 2018

zbMATH: 1395.60001
MathSciNet: MR3790069
Digital Object Identifier: 10.1214/17-PS295

Primary: 11-02 , 60-01
Secondary: 11K45 , 60F15 , 65C10

Keywords: $\infty$-distributed , completely equidistributed , Completely uniformly distributed , Dependent random variables , equidistribution , metric theory , pseudo-random numbers , Strong law of large numbers , weakly completely uniformly distributed , weakly correlated , Weyl criterion

Vol.15 • 2018
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