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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 absolutelycontinuous and the discrete components of $F$. In this paper we study specialcases of models, including stationary Markov chains of any order and stationaryrenewal point processes, where we establish a law of pure types: $F$ is theneither a uniform or a singular CDF on $[0,1]$. Moreover, we study mixtures ofsuch models. In most cases expressions and plots of $F$ are given.

Singular distribution functions for random variables with stationary digits