# Characterization of random variables with stationary digits

Journal of Applied Probability

Published On 2022/12

Let be an integer, a stochastic process with state space , and F the cumulative distribution function (CDF) of . We show that stationarity of 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 is a Rajchman measure if and only if F is the uniform CDF on [0, 1].

Journal

Journal of Applied Probability

Published On

2022/12

Volume

59

Issue

4

Page

931-947

## Authors

#### Jesper Møller

##### Aalborg Universitet

Position

Professor in Statistics

H-Index(all)

46

H-Index(since 2020)

23

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

Mathematical Statistics

Probability Theory

University Profile Page

#### Ira Herbst

##### University of Virginia

Position

Professor of Mathematics

H-Index(all)

36

H-Index(since 2020)

14

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

Mathematical Physics

University Profile Page

#### Horia D. Cornean

##### Aalborg Universitet

Position

Professor of Mathematics Denmark

H-Index(all)

22

H-Index(since 2020)

13

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

spectral theory

scattering theory

quantum transport

ICT

University Profile Page

### Other Articles from authors

Ira Herbst

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##### Color-avoiding percolation and branching processes

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