Characterization of random variables with stationary digits

Horia D. Cornean, Ira W. Herbst, Jesper Møller, Kasper S. Sørensen, Benjamin B. Støttrup

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningpeer review


Let q ≥ 2 be an integer, {Xn}n≥1 a stochastic process with state space {0, . . ., q − 1}, and F the cumulative distribution function (CDF) of Σ∞n=1 Xnq−n. We show that stationarity of {Xn}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].

TidsskriftJournal of Applied Probability
Udgave nummer4
Sider (fra-til)931-947
Antal sider17
StatusUdgivet - 15 dec. 2022

Bibliografisk note

Funding Information:
JM and HC are supported in part by The Danish Council for Independent Research | Natural Sciences, grant 7014-00074B, ‘Statistics for point processes in space and beyond’. JM, BS, and KSS are supported in part by the Centre for Stochastic Geometry and Advanced Bioimaging, funded by grant 8721 from the Villum Foundation. This work was supported by The Danish Council for Independent Research | Natural Sciences, grant DFF – 10.46540/2032-00005B.

Publisher Copyright:
© The Author(s), 2022.


Dyk ned i forskningsemnerne om 'Characterization of random variables with stationary digits'. Sammen danner de et unikt fingeraftryk.