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Coupling results and Markovian structures for number representations of continuous random variables

Part of: Stochastic processes

Published online by Cambridge University Press:  28 April 2025

Jesper Møller Open the ORCID record for Jesper Møller [Opens in a new window]
Jesper Møller*
Affiliation:
Aalborg University
*
*Postal address: Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark. Email: jm@math.aau.dk
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Abstract

Consider nested subdivisions of a bounded real set into intervals defining the digits $X_1,X_2,\ldots$ of a random variable X with a probability density function f. If f is almost everywhere lower semi-continuous, there is a non-negative integer-valued random variable N such that the distribution of $R=(X_{N+1},X_{N+2},\ldots)$ conditioned on $S=(X_1,\ldots,X_N)$ does not depend on f. If also the lengths of the intervals exhibit a Markovian structure, $R\mid S$ becomes a Markov chain of a certain order $s\ge0$. If $s=0$ then $X_{N+1},X_{N+2},\ldots$ are independent and identically distributed with a known distribution. When $s>0$ and the Markov chain is uniformly geometric ergodic, there is a random time M such that the chain after time $\max\{N,s\}+M-s$ is stationary and M follows a simple known distribution.

Keywords

Approximationbeta-expansioncontinued fractiondynamical systemLüroth seriesMarkov chainnested partitionspseudo golden ratioread-once algorithmsufficient digits

MSC classification

Primary: 60G07: General theory of processes
Secondary: 60G99: None of the above, but in this section

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 20
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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