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On the asymptotic behavior of the Diaconis–Freedman chain in a multi-dimensional simplex

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  28 January 2022

Marc Peigné  and
Tat Dat Tran
Marc Peigné*
Affiliation:
Institut Denis Poisson, Université de Tours
Tat Dat Tran*
Affiliation:
Max-Planck-Institut für Mathematik in den Naturwissenschaften Mathematisches Institut, Universität Leipzig
*
*Postal address: Institut Denis Poisson UMR 7013, Université de Tours, Université d’Orléans, CNRS France. Email: peigne@univ-tours.fr
**Postal address: Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22, D-04103 Leipzig, Germany. Email: trandat@mis.mpg.de
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Abstract

We give a setting of the Diaconis–Freedman chain in a multi-dimensional simplex and consider its asymptotic behavior. By using techniques from random iterated function theory and quasi-compact operator theory, we first give some sufficient conditions which ensure the existence and uniqueness of an invariant probability measure and, in particular cases, explicit formulas for the invariant probability density. Moreover, we completely classify all behaviors of this chain in dimension two. Some other settings of the chain are also discussed.

Keywords

Iterated functions systemsquasi-compact linear operatorsabsorbing compact setinvariant probability measureinvariant probability density

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60F05: Central limit and other weak theorems
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 2 , June 2022 , pp. 505 - 526
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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