Characterization of random variables with stationary digits

Abstract

Let $q\ge 2$ be an integer, $\{X_n{\}}_{n\ge 1}$ a stochastic process with state space $\{0,\dots ,q-1\}$, and $F$ the cumulative distribution function (CDF) of ${\sum }_{n=1}^{\infty }{X}_n{q}^{-n}$. We show that stationarity of $\{X_n{\}}_{n\ge 1}$ is equivalent to a functional equation obeyed by $F$ and use this to characterize the characteristic function of $X$ and the structure of $F$ in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of $F$ can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that $\mathrm{d}F$ is a Rajchman measure if and only if $F$ is the uniform CDF on $[0,1]$.

20 pages and 1 figure