  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 of $\{X_n\}_{n\geq
1}$ 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 $[0,1]$.
