Hostname: page-component-669899f699-vbsjw Total loading time: 0 Render date: 2025-05-01T17:38:49.422Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Barndorff-Nielsen, O. (1978). Information and Exponential Families in Statistical Theory. John Wiley, Chichester.Google Scholar
Barrionuevo, J., Burton, R., Dajani, K. and Kraiikamp, C. (1996). Ergodic properties of generalized Lüroth series. Acta Arith. 74, 311–327.CrossRefGoogle Scholar
Beraha, M. and Møller, J. (2025). Generalized finite Polya tree distributions for statistical modelling. In preparation.Google Scholar
Cornean, H., Herbst, I., Møller, J., Støttrup, B. and Studsgaard, K. (2022). Characterization of random variables with stationary digits. J. Appl. Prob. 59, 931–947.CrossRefGoogle Scholar
Cornean, H., Herbst, I., Møller, J., Støttrup, B. and Studsgaard, K. (2023). Singular distribution functions for random variables with stationary digits. Methodol. Comput. Appl. Prob. 25, 31.CrossRefGoogle Scholar
Dajani, K. and Kalle, C. (2021). A First Course in Ergodic Theory. Chapman and Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
Dajani, K. and Kraaikamp, C. (2002). Ergodic Theory of Numbers. Mathematical Association of America, Washington, DC.CrossRefGoogle Scholar
Daley, D. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, Vol. I, Elementary Theory and Methods. Springer, New York.Google Scholar
Foss, S. and Tweedie, R. (1998). Perfect simulation and backward coupling. Stoch. Models 14, 187–203.CrossRefGoogle Scholar
Gauss, C. (1976). Mathematisches Tagebuch 1796–1814. Akademische Verlagsgesellschaft, Geest & Portig K.G., Leipzig.Google Scholar
Gelfond, A. (1959). A common property of number systems. Izv. Akad. Nauk. SSSR. Ser. Mat. 23, 809–814.Google Scholar
Herbst, I., Møller, J. and Svane, A. (2023). The asymptotic distribution of the remainder in a certain base- $\beta$ expansion. Preprint, arXiv:2312.09652.Google Scholar
Herbst, I., Møller, J. and Svane, A. (2023). How many digits are needed? Preprint, arXiv:2307.06685.Google Scholar
Herbst, I., Møller, J. and Svane, A. (2025). The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable. To appear in Methodol. Comput. Appl. Prob. 27.CrossRefGoogle Scholar
Herbst, I. W., Møller, J. and Svane, A. M. (2024). How many digits are needed? Methodol. Comput. Appl. Prob. 26, 5.CrossRefGoogle Scholar
Khintchine, A. (1963). Continued Fractions. Noordhoff, Groningen.Google Scholar
Lavine, M. (1992). Some aspects of Polya tree distributions for statistical modelling. Ann. Statist. 20, 1222–1235.CrossRefGoogle Scholar
Lavine, M. (1994). More aspects of Polya tree distributions for statistical modelling. Ann. Statist. 22, 1161–1176.CrossRefGoogle Scholar
Lüroth, J. (1883). Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe. Math. Annalen 21, 411–423.CrossRefGoogle Scholar
Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle Scholar
Møller, J. and Waagepetersen, R. (2004). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton, FL.Google Scholar
Parry, W. (1960). On the $\beta$ -expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11, 401–416.CrossRefGoogle Scholar
Rényi, A. (1957). Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8, 477–493.CrossRefGoogle Scholar
Schweiger, F. (1995). Ergodic Theory of Fibered Systems and Metric Number Theory. Clarendon Press, Oxford.Google Scholar
Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.CrossRefGoogle Scholar
Wilson, D. (2000). How to couple from the past using a read-once source of randomness. Random Structures Algorithms 16, 85–113.3.0.CO;2-H>CrossRefGoogle Scholar
Wong, W. and Ma, L. (2010). Optional Pólya tree and Bayesian inference. Ann. Statist. 38, 1433–1459.CrossRefGoogle Scholar