Characterization of random variables with stationary digits

Let q ≥ 2 be an integer, {X_n}_n≥1 a stochastic process with state space {0, . . ., q − 1}, and F the cumulative distribution function (CDF) of ^Σ∞_n=1 X_nq⁻ⁿ. We show that stationarity of {X_n}_n≥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 dF is a Rajchman measure if and only if F is the uniform CDF on [0, 1].