Singular Distribution Functions for Random Variables with Stationary Digits

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


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Let F be the cumulative distribution function (CDF) of the base-q expansion ∑n=1∞Xnq-n, where q≥ 2 is an integer and {Xn}n≥1 is a stationary stochastic process with state space { 0 , … , q- 1 }. In a previous paper we characterized the absolutely continuous and the discrete components of F. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: F is then either a uniform or a singular CDF on [0, 1]. Moreover, we study mixtures of such models. In most cases expressions and plots of F are given.

TidsskriftMethodology and Computing in Applied Probability
Udgave nummer1
StatusUdgivet - mar. 2023

Bibliografisk note

Funding Information:
This work was supported by The Danish Council for Independent Research | Natural Sciences, grant DFF - 10.46540/2032-00005B.

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.


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