Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T16:19:16.245Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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
Check for updates
Copyright
© The Author(s), 2022. 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.)

References

Barnsley, M. F., Demko, S. G., Elton, J. H. and Geronimo, J. S. (1988). Invariant measures for Markov processes arising from iterated functions systems with place-dependent probabilities. Ann. Inst. H. Poincaré Prob. Statist. 24, 367–394.Google Scholar
Barnsley, M. F. and Elton, J. H. (1988). A new class of Markov processes for image encoding. Adv. Appl. Prob. 20, 14–32.CrossRefGoogle Scholar
Barnsley, M. F., Elton, J. H. and Hardin, D. P. (1989). Recurrent iterated function systems. Constr. Approx. 5, 3–31.CrossRefGoogle Scholar
Borovkov, A. A. (1998). Ergodicity and Stability of Stochastic Processes. Wiley, New York.Google Scholar
Bush, R. R. and Mosteller, F. (1953). A stochastic model with applications to learning. Ann. Math. Statist. 24, 559–585.CrossRefGoogle Scholar
DeGroot, M. H.and Rao, M. M. (1963). Stochastic give-and-take. J. Math. Anal. Appl. 7, 489–498.CrossRefGoogle Scholar
Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41, 45–76.CrossRefGoogle Scholar
Dubins, L. E. and Freedman, D. A. (1966). Invariant probabilities for certain Markov processes. Ann. Math. Statist. 37, 837–848.CrossRefGoogle Scholar
Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1, 209–230.CrossRefGoogle Scholar
Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer, New York.Google Scholar
Guivarc’h, Y. and Raugi, A. (1986). Products of random matrices: convergence theorems. In Random Matrices and their Applications (Contemp. Math. 50), eds. Cohen, J. E., Kesten, H. and Newman, C. M.. American Mathematical Society, Providence, RI, pp. 31–54.Google Scholar
Hennion, H. (1993). Sur un théorème spectral et son application aux noyaux lipchitziens. Proc. Amer. Math. Soc. 118, 627–634.Google Scholar
Hennion, H. and Hervé, L. c. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness (Lect. Notes Math. 1766). Springer, Berlin.Google Scholar
Hervé, L. c. (1994). Étude d’opérateurs quasi-compacts positifs. Applications aux opérateurs de transfert. Ann. Inst. H. Poincaré Prob. Statist. 30, 437–466.Google Scholar
Hofrichter, J., Jost, J., and Tran, T. D. (2017). Information Geometry and Population Genetics. Springer, Cham.CrossRefGoogle Scholar
Jost, J., Le, H., and Tran, T. (2021). Probabilistic morphisms and Bayesian nonparametrics. Eur. Phys. J. Plus 136, 441.CrossRefGoogle Scholar
Kaijser, T. (1981). On a new contraction condition for random systems with complete connections. Rev. Roumaine Math. Pure Appl. 26, 1075–1117.Google Scholar
Kapica, R. and Sleczka, M. (2017). Random iteration with place-dependent probabilities. Preprint, arXiv:1107.0707.Google Scholar
Karlin, S. (1953). Some random walks arising in learning models. I. Pacific J. Math. 3, 725–756.CrossRefGoogle Scholar
Ladjimi, F. and Peigné, M. (2019). On the asymptotic behavior of the Diaconis and Freedman chain on [0, 1]. Statist. Prob. Lett. 145, 1–11.CrossRefGoogle Scholar
Letac, G. (1986). A contraction principle for certain Markov chains and its applications. In Random Matrices and their Applications (Contemp. Math. 50), eds. J. E. Cohen, H. Kesten and C. M. Newman. American Mathematical Society, Providence, RI, pp. 263–273.Google Scholar
Mirek, M. (2011). Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps. Prob. Theory Relat. Fields 151, 705–734.CrossRefGoogle Scholar
Nguyen, T.-M. and Volkov, S. (2020). On a class of random walks in simplexes. J. Appl. Prob. 57, 409–428.CrossRefGoogle Scholar
Pacheco-Gonzalez, C. G. and Stoyanov, J. (2008) A class of Markov chains with beta ergodic distribution. Math. Scientist 33, 110–119.Google Scholar
Peigné, M. (1993). Iterated function systems and spectral decomposition of the associated Markov operator. Publications mathématiques et informatique de Rennes 2, 1–28.Google Scholar
Peigné, M. and Woess, W. (2011a). Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity. Colloq. Math. 125, 31–54.Google Scholar
Peigné, M. and Woess, W. (2011b). Stochastic dynamical systems with weak contractivity properties II. Iteration of Lipschitz mappings. Colloq. Math. 125, 55–81.Google Scholar
Pirinskyb, C. and Stoyanov, J. (2000) Random motions, classes of ergodic Markov chains and beta distributions. Statist. Prob. Lett. 50, 293–304.Google Scholar
Ramli, M. A. and Leng, G. (2010). The stationary probability density of a class of bounded Markov processes. Adv. Appl. Prob. 42, 986–993.CrossRefGoogle Scholar
Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica 4, 639–650.Google Scholar
Stenflo, O. (2012). A survey of average contractive iterated function systems. J. Difference Equat. Appl. 18, 1355–1380.CrossRefGoogle Scholar
Tran, T., Hofrichter, J. and Jost, J. (2015a). The free energy method and the Wright–Fisher model with 2 alleles. Theory Biosci. 134, 83–92.CrossRefGoogle ScholarPubMed
Tran, T., Hofrichter, J. and Jost, J. (2015b). The free energy method for the Fokker–Planck equation of the Wright–Fisher model. MIS-Preprint 29/2015.Google Scholar