Transfer of stationary stochastic processes to the unit interval. Part I: Characterization results.
Horia D. Cornean, Ira Herbst, Jesper Michael Moller, Benjamin B. Støttrup, Kasper S. Sørensen · VBN Forskningsportal (Aalborg Universitet) · 2020
Let $q\ge2$ be an integer and let $\{X_n\}_{n\geq 1}$ be a stochastic process with state space $\{0,\ldots,q-1\}$. Let $F$ be the cumulative distribution function (CDF) of the base-$q$ expansion $\sum_{n=1}^\infty X_n q^{-n}$. We show that stationarity of $\{X_n\}_{n\geq 1}$ is equivalent to a functional equation obeyed by $F$. Using this equation we characterize the structure of $F$ in terms of its Lebesgue decomposition. More precisely, we prove that the absolutely continuous component of $F$ can only be the uniform distribution on the unit interval, while its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also characterize the stationarity of $\{X_n\}_{n\geq 1}$ in terms of the characteristic function of $\mathrm d F$. In particular, we show that $\mathrm d F$ is a Rajchman measure if and only if $F $ is the uniform CDF on $[0,1]$. Hence $F$ cannot be Minkowski's question-mark function restricted to $[0,1]$.