Singular Distribution Functions for Random Variables with Stationary Digits

Let F be the cumulative distribution function (CDF) of the base-q expansion n-expressionry sumexpressiontion (infinity)(n=1)X(n)q(-n) , where q >= 2 is an integer and {X-n}(n >= 1 )is a stationary stochastic process with state space {0,...,q-1} . In a previous paper we characterized the absolutely continuous and the dis-crete 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