Singular distribution functions for random variables with stationary digits

TL;DR

This paper investigates the distribution functions of random variables with stationary digits in base-$q$ expansions, establishing conditions under which these functions are either uniform or singular, and analyzing mixtures of models.

Contribution

It characterizes the distribution functions for stationary digit models, showing they are either uniform or singular, and extends analysis to mixtures of such models.

Findings

$F$ is either a uniform or singular distribution function.

Established a law of pure types for stationary digit models.

Provided explicit expressions and plots for $F$ in various cases.

Abstract

Let $F$ be the cumulative distribution function (CDF) of the base-$q$ expansion $\sum_{n=1}^\infty X_n q^{-n}$, where $q\ge2$ is an integer and $\{X_n\}_{n\geq 1}$ is a stationary stochastic process with state space $\{0,\ldots,q-1\}$. In a previous paper we characterized the absolutely continuous and the discrete components of $F$. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: $F$ is then either a uniform or a singular CDF on $[0,1]$. Moreover, we study mixtures of such models. In most cases expressions and plots of $F$ are given.