The theory of equidistribution is about hundred years old, and has beendeveloped primarily by number theorists and theoretical computer scientists. Amotivated uninitiated peer could encounter difficulties perusing theliterature, 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 secondauthor from 2002. Using it, various new examples of completely uniformlydistributed (mod 1) sequences, in the "metric" (meaning almost sure stochastic)sense, can be easily exhibited. In particular, we point out naturalgeneralizations 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 Weylin 1916. In passing, we also derive a Weyl-like criterion for weakly completelyequidistributed (also known as WCUD) sequences, of substantial recent interestin MCMC simulations. The translation from number theory to probability language brings into focusa version of the strong law of large numbers for weakly correlatedcomplex-valued random variables, the study of which was initiated by Weyl inthe aforementioned manuscript, followed up by Davenport, Erd\"{o}s and LeVequein 1963, and greatly extended by Russell Lyons in 1988. In this context, anapplication 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 byNiederreiter 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 ofconsiderable interest to probabilists and statisticians, as well as certaincomputer scientists and number theorists.
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机译：公平分配理论已有一百多年的历史，并且主要是由数论家和理论计算机科学家开发的。由于不同学校使用不同的同义词和多义词，有积极主动的同伴可能会在阅读文学时遇到困难。本说明的目的是为概率论者提供简短的介绍。我们首先回顾一下源于2002年第二作者的工作的观点。使用它，可以很容易地展示“度量”（几乎肯定是随机的）意义上的完全均匀分布（mod 1）序列的各种新示例。特别是，我们指出原始的$ p $乘以等分布序列$ k ^ p \，t $ mod 1，$ k \ geq 1 $（其中$ p \ in \ mathbb {N} $和$ t \ in [0,1] $），这要归功于赫尔曼·威林（Hermann Weylin）1916年。顺便说一句，我们还为最近在MCMC模拟中引起关注的弱完全等配（也称为WCUD）序列推导了类似Weyl的准则。从数论到概率语言的转换成为弱相关复数值随机变量的大数强定律的焦点版本，该研究由Weyl在上述手稿中发起，随后是Davenport，Erd \“ {o} s和LeVequein 1963，并在1988年由Russell Lyons大大扩展。在这种情况下，对$ \ infty-分布的Koksma数$ t ^ k $ mod 1，$ k \ geq 1 $（其中$ t \ in [1 ，a] $代表$ a> 1 $），并讨论了Niederreiter和Tichy自1985年以来的重要概括，其中严格意义上讲，新数学的数量可忽略不计，但其观点和最终提出的开放性问题可能是对概率论者和统计学家以及某些计算机科学家和数论学家的兴趣非常大。
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