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Characterization of random variables with stationary digits

Part of: Stochastic processes

Published online by Cambridge University Press:  15 August 2022

Horia D. Cornean Open the ORCID record for Horia D. Cornean [Opens in a new window] ,
Ira W. Herbst ,
Jesper Møller Open the ORCID record for Jesper Møller [Opens in a new window] ,
Kasper S. Sørensen  and
Benjamin B. Støttrup Open the ORCID record for Benjamin B. Støttrup [Opens in a new window]
Horia D. Cornean*
Affiliation:
Aalborg University
Ira W. Herbst*
Affiliation:
University of Virginia
Jesper Møller*
Affiliation:
Aalborg University
Kasper S. Sørensen*
Affiliation:
Aalborg University
Benjamin B. Støttrup*
Affiliation:
Aalborg University
*
*Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark
**Postal address: Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA
*Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark
*Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark
*Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark
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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 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].

Keywords

Functional equationLebesgue decompositionMinkowski’s question-mark functionmixture distributionRajchman measuresingular function

MSC classification

Primary: 60G10: Stationary processes
Secondary: 60G30: Continuity and singularity of induced measures
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 931 - 947
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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