Characterization of random variables with stationary digits

Abstract Let $q\ge2$ be an integer, $\{X_n\}_{n\geq 1}$ a stochastic process with state space $\{0,\ldots,q-1\}$ , and F the cumulative distribution function (CDF) of $\sum_{n=1}^\infty X_n q^{-n}$ . We show that stationarity is equivalent to functional equation obeyed by use this characterize characteristic X structure in terms its Lebesgue decomposition. More precisely, while absolutely continuous component can only uniform on unit interval, discrete countable convex combination certain explicitly computable CDFs for probability distributions finite support. also $\mathrm{d} F$ Rajchman measure if CDF [0, 1].