Abstract
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].
Original language | English |
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Journal | Journal of Applied Probability |
Volume | 59 |
Issue number | 4 |
Pages (from-to) | 931-947 |
Number of pages | 17 |
ISSN | 0021-9002 |
DOIs | |
Publication status | Published - 15 Dec 2022 |
Bibliographical note
Publisher Copyright:© The Author(s), 2022.
Keywords
- Functional equation
- Lebesgue decomposition
- Minkowski’s question-mark function
- mixture distribution
- Rajchman measure
- singular function