Singular Distribution Functions for Random Variables with Stationary Digits

Abstract

Let F be the cumulative distribution function (CDF) of the base-q expansion $$\sum _{n=1}^\infty X_n q^{-n}$$ , where $$q\ge 2$$ is an integer and $$\{X_n\}_{n\ge 1}$$ 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.