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Robustness of iterated function systems of Lipschitz maps

Part of: Parametric inference Special classes of linear operators Markov processes

Published online by Cambridge University Press:  28 February 2023

Loïc Hervé  and
James Ledoux Open the ORCID record for James Ledoux [Opens in a new window]
Loïc Hervé*
Affiliation:
Université de Rennes, INSA Rennes, CNRS, IRMAR-UMR 6625
James Ledoux*
Affiliation:
Université de Rennes, INSA Rennes, CNRS, IRMAR-UMR 6625
*
*Postal address: INSA Rennes, 20 Avenue des Buttes de Cöesmes, CS 70 839, 35708 Rennes Cedex 7, France.
*Postal address: INSA Rennes, 20 Avenue des Buttes de Cöesmes, CS 70 839, 35708 Rennes Cedex 7, France.
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Abstract

Let $\{X_n\}_{n\in{\mathbb{N}}}$ be an ${\mathbb{X}}$-valued iterated function system (IFS) of Lipschitz maps defined as $X_0 \in {\mathbb{X}}$ and for $n\geq 1$, $X_n\;:\!=\;F(X_{n-1},\vartheta_n)$, where $\{\vartheta_n\}_{n \ge 1}$ are independent and identically distributed random variables with common probability distribution $\mathfrak{p}$, $F(\cdot,\cdot)$ is Lipschitz continuous in the first variable, and $X_0$ is independent of $\{\vartheta_n\}_{n \ge 1}$. Under parametric perturbation of both F and $\mathfrak{p}$, we are interested in the robustness of the V-geometrical ergodicity property of $\{X_n\}_{n\in{\mathbb{N}}}$, of its invariant probability measure, and finally of the probability distribution of $X_n$. Specifically, we propose a pattern of assumptions for studying such robustness properties for an IFS. This pattern is implemented for the autoregressive processes with autoregressive conditional heteroscedastic errors, and for IFS under roundoff error or under thresholding/truncation. Moreover, we provide a general set of assumptions covering the classical Feller-type hypotheses for an IFS to be a V-geometrical ergodic process. An accurate bound for the rate of convergence is also provided.

Keywords

Markov chaingeometric ergodicityperturbationnon-linear stochastic recursionautoregressive process

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 47B07: Operators defined by compactness properties 62F99: None of the above, but in this section
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 921 - 944
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Alquier, P., Friel, N., Everitt, R. and Boland, A. (2016). Noisy Monte Carlo: Convergence of Markov chains with approximate transition kernels. Statist. Comput. 26, 29–47.CrossRefGoogle Scholar
Alsmeyer, G. (2003). On the Harris recurrence of iterated random Lipschitz functions and related convergence rate results. J. Theoret. Prob. 16, 217–247.10.1023/A:1022290807360CrossRefGoogle Scholar
Altman, E., Avrachenkov, K. E. and Núñez-Queija, R. (2004). Perturbation analysis for denumerable Markov chains with application to queueing models. Adv. Appl. Prob. 36, 839–853.10.1239/aap/1093962237CrossRefGoogle Scholar
Baladi, V. (2000). Positive Transfer Operators and Decay of Correlations (Adv. Ser. Nonlinear Dynamics 16). World Scientific, River Edge, NJ.CrossRefGoogle Scholar
Baxendale, P. H. (2005). Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Prob. 15, 700–738.10.1214/105051604000000710CrossRefGoogle Scholar
Benda, M. (1998). A central limit theorem for contractive stochastic dynamical systems. J. Appl. Prob. 35, 200–205.CrossRefGoogle Scholar
Breyer, L., Roberts, G. O. and Rosenthal, J. S. (2001). A note on geometric ergodicity and floating-point roundoff error. Statist. Prob. Lett. 53, 123–127.CrossRefGoogle Scholar
Cline, D. B. H. (2007). Regular variation of order 1 nonlinear AR-ARCH models. Stoch. Process. Appl. 117, 840–861.10.1016/j.spa.2006.10.009CrossRefGoogle Scholar
Cline, D. B. H. (2007). Stability of nonlinear stochastic recursions with application to nonlinear AR-GARCH models. Adv. Appl. Prob. 39, 462–491.CrossRefGoogle Scholar
Cline, D. B. H. and Pu, H. H. (2002). A note on a simple Markov bilinear stochastic process. Statist. Prob. Lett. 56, 283–288.CrossRefGoogle Scholar
Douc, R. and Moulines, E. (2012). Asymptotic properties of the maximum likelihood estimation in misspecified hidden Markov models. Ann. Statist. 40, 2697–2732.10.1214/12-AOS1047CrossRefGoogle Scholar
Douc, R., Moulines, E., Priouret, P. and Soulier, P. (2018). Markov Chains. Springer, New York.CrossRefGoogle Scholar
Duflo, M. (1997). Random Iterative Models. Springer, Berlin.Google Scholar
Ferré, D., Hervé, L. and Ledoux, J. (2013). Regular perturbation of V-geometrically ergodic Markov chains. J. Appl. Prob. 50, 184–194.CrossRefGoogle Scholar
Galtchouk, L. and Pergamenshchikov, S. (2014). Geometric ergodicity for classes of homogeneous Markov chains. Stoch. Process. Appl. 124, 3362–3391.CrossRefGoogle Scholar
Gouëzel, S. and Liverani, C. (2006). Banach spaces adapted to Anosov systems. Ergodic Theory Dynam. Systems 26, 189–217.CrossRefGoogle Scholar
Greenwood, P. E. and Wefelmeyer, W. (1998). Maximum likelihood estimator and Kullback–Leibler information in misspecified Markov chain models. Theory Prob. Appl. 42, 103–111.CrossRefGoogle Scholar
Guibourg, D., Hervé, L. and Ledoux, J. (2011). Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity. Preprint, arXiv:1110.3240.Google Scholar
Hairer, M. and Mattingly, J. C. (2011). Yet another look at Harris’ ergodic theorem for Markov chains. In Seminar on Stochastic Analysis, Random Fields and Applications VI, ed. R. Dalang, M. Dozzi and F. Russo. Springer, Basel, pp. 109–117.CrossRefGoogle 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. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness (Lect. Notes Math. 1766). Springer, New York.CrossRefGoogle Scholar
Hervé, L. (2008). Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces. Ann. Inst. H. Poincaré Prob. Statist. 44, 1090–1095.CrossRefGoogle Scholar
Hervé, L. and Ledoux, J. (2014). Approximating Markov chains and V-geometric ergodicity via weak perturbation theory. Stoch. Process. Appl. 124, 613–638.10.1016/j.spa.2013.09.003CrossRefGoogle Scholar
Hervé, L. and Ledoux, J. (2014). Spectral analysis of Markov kernels and aplication to the convergence rate of discrete random walks. Adv. Appl. Prob. 46, 1036–1058.CrossRefGoogle Scholar
Johndrow, J. E., Mattingly, J. C., Mukherjee, S. and Dunson, D. Optimal approximating Markov chains for Bayesian inference. Preprint, arXiv:1508.03387v3.Google Scholar
Kartashov, N. V. (1986). Inequalities in theorems of ergodicity and stability for Markov chains with common phase space I. Theory Prob. Appl. 30, 247–259.CrossRefGoogle Scholar
Kartashov, N. V. (1986). Inequalities in theorems of ergodicity and stability for Markov chains with common phase space II. Theory Prob. Appl. 30, 505–515.Google Scholar
Keller, G. (1982). Stochastic stability in some chaotic dynamical systems. Monatshefte Math. 94, 313–333.10.1007/BF01667385CrossRefGoogle Scholar
Keller, G. and Liverani, C. (1999). Stability of the spectrum for transfer operators. Annali della Scuola Normale Superiore di Pisa – Classe di Scienze (4) XXVIII, 141–152.Google Scholar
Le Page, É. (1983). Théorèmes de renouvellement pour les produits de matrices aléatoires. Équations aux différences aléatoires. In Séminaires de probabilités Rennes 1983. Publ. Sém. Math. Univ. Rennes I, Rennes, p. 116.Google Scholar
Liverani, C. (2004). Invariant measures and their properties. A functional analytic point of view. In Dynamical Systems. Part II: Topological Geometrical and Ergodic Properties of Dynamical Systems, ed. S. Marmi (Publ. Scuola Normale Superiore). Springer, New York.Google Scholar
Lund, R. B. and Tweedie, R. L. (1996). Geometric convergence rates for stochastically ordered Markov chains. Math. Operat. Res. 21, 182–194.10.1287/moor.21.1.182CrossRefGoogle Scholar
Medina-Aguayo, F. J., Lee, A. and Roberts, G. O. (2016). Stability of noisy Metropolis–Hastings. Statist. Comput. 26, 1187–1211.CrossRefGoogle ScholarPubMed
Medina-Aguayo, F. J., Rudolf, D. and Schweizer, N. (2020). Perturbation bounds for Monte Carlo within Metropolis via restricted approximations. Stoch. Process. Appl. 130, 2200–2227.10.1016/j.spa.2019.06.015CrossRefGoogle ScholarPubMed
Meitz, M. and Saikkonen, P. (2010). A note on the geometric ergodicity of a nonlinear AR-ARCH model. Statist. Prob. Lett. 80, 631–638.10.1016/j.spl.2009.12.020CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.10.1007/978-1-4471-3267-7CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Prob. 4, 981–1011.Google Scholar
Milhaud, X. and Raugi, A. (1989). Etude de l’estimateur du maximum de vraisemblance dans le cas d’un processus autorégressif : convergence, normalité asymptotique, vitesse de convergence. Ann. Inst. H. Poincaré Prob. Statist. 25, 383–428.Google Scholar
Mitrophanov, A. Y. (2005). Sensitivity and convergence of uniformly ergodic Markov chains. J. Appl. Prob. 42, 1003–1014.CrossRefGoogle Scholar
Mouhoubi, Z. and Assani, D. (2010). New perturbation bounds for denumerable Markov chains. Linear Algebra Appl. 432, 1627–1649.CrossRefGoogle Scholar
Negrea, J. and Rosenthal, J. S. Approximations of geometrically ergodic reversible Markov chains. Adv. Appl. Prob. 53, 981–1022.10.1017/apr.2021.10CrossRefGoogle Scholar
Roberts, G. O., Rosenthal, J. S. and Schwartz, P. O. (1998). Convergence properties of perturbed Markov chains. J. Appl. Prob. 35, 1–11.10.1239/jap/1032192546CrossRefGoogle Scholar
Rudolf, D. and Schweizer, N. (2018). Perturbation theory for Markov chains via Wasserstein distance. Bernoulli 24, 2610–2639.CrossRefGoogle Scholar
Scheffé, H. (1947). A useful convergence theorem for probability distributions. Ann. Math. Statist. 18, 434–438.CrossRefGoogle Scholar
Shardlow, T. and Stuart, A. M. (2000). A perturbation theory for ergodic Markov chains and application to numerical approximations. SIAM J. Numer. Anal. 37, 1120–1137.CrossRefGoogle Scholar
Tsay, R. S. (2010). Analysis of Financial Time Series, 3rd edn. John Wiley, Hoboken, NJ.CrossRefGoogle Scholar
Wu, L. (2004). Essential spectral radius for Markov semigroups. I. Discrete time case. Prob. Theory Relat. Fields 128, 255–321.CrossRefGoogle Scholar