Bernoulli

  • Bernoulli
  • Volume 23, Number 4A (2017), 2129-2180.

Convergence rate of the powers of an operator. Applications to stochastic systems

Bernard Delyon

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Abstract

We extend the traditional operator theoretic approach for the study of dynamical systems in order to handle the problem of non-geometric convergence. We show that the probabilistic treatment developed and popularized under Richard Tweedie’s impulsion, can be placed into an operator framework in the spirit of Yosida–Kakutani’s approach. General theorems as well as specific results for Markov chains are given. Application examples to general classes of Markov chains and dynamical systems are presented.

Article information

Source
Bernoulli, Volume 23, Number 4A (2017), 2129-2180.

Dates
Received: May 2014
Revised: August 2015
First available in Project Euclid: 9 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1494316815

Digital Object Identifier
doi:10.3150/15-BEJ778

Mathematical Reviews number (MathSciNet)
MR3648028

Zentralblatt MATH identifier
06778239

Keywords
Markov chains

Citation

Delyon, Bernard. Convergence rate of the powers of an operator. Applications to stochastic systems. Bernoulli 23 (2017), no. 4A, 2129--2180. doi:10.3150/15-BEJ778. https://projecteuclid.org/euclid.bj/1494316815


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References

  • [1] Aubin, J.-P. and Cellina, A. (1984). Differential Inclusions: Set-Valued Maps and Viability Theory. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 264. Berlin: Springer.
  • [2] Butkovsky, O. (2014). Subgeometric rates of convergence of Markov processes in the Wasserstein metric. Ann. Appl. Probab. 24 526–552.
  • [3] Cloez, B. and Hairer, M. (2015). Exponential ergodicity for Markov processes with random switching. Bernoulli 21 505–536.
  • [4] de Leeuw, K. and Glicksberg, I. (1961). Applications of almost periodic compactifications. Acta Math. 105 63–97.
  • [5] Doeblin, W. and Fortet, R. (1937). Sur des chaînes à liaisons complètes. Bull. Soc. Math. France 65 132–148.
  • [6] Douc, R., Fort, G., Moulines, E. and Soulier, P. (2004). Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14 1353–1377.
  • [7] Douc, R., Moulines, E. and Soulier, P. (2007). Computable convergence rates for sub-geometric ergodic Markov chains. Bernoulli 13 831–848.
  • [8] Dunford, N. and Schwartz, J.T. (1988). Linear operators. Part I. In General Theory, with the Assistance of. Wiley Classics Library. New York: Wiley.
  • [9] Durmus, A., Fort, G. and Moulines, É. (2016). Subgeometric rates of convergence in Wasserstein distance for Markov chains. Ann. Inst. Henri Poincaré Probab. Stat. 52 1799–1822.
  • [10] Gantmacher, F.R. (1998). The Theory of Matrices. Vol. 1. Providence, RI: AMS Chelsea Publishing.
  • [11] Gouëzel, S. (2004). Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128 82–122.
  • [12] Hennion, H. and Hervé, L. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Berlin: Springer.
  • [13] Ionescu Tulcea, C.T. and Marinescu, G. (1950). Théorie ergodique pour des classes d’opérations non complètement continues. Ann. of Math. (2) 52 140–147.
  • [14] Iosifescu, M. (1993). A basic tool in mathematical chaos theory: Doeblin and Fortet’s ergodic theorem and Ionescu Tulcea and Marinescu’s generalization. In Doeblin and Modern Probability (Blaubeuren, 1991). Contemp. Math. 149 111–124. Providence, RI: Amer. Math. Soc.
  • [15] Iosifescu, M. and Theodorescu, R. (1969). Random Processes and Learning. New York: Springer.
  • [16] Jarner, S.F. and Roberts, G.O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Probab. 12 224–247.
  • [17] Joulin, A. and Ollivier, Y. (2010). Curvature, concentration and error estimates for Markov chain Monte Carlo. Ann. Probab. 38 2418–2442.
  • [18] Kontoyiannis, I. and Meyn, S.P. (2003). Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab. 13 304–362.
  • [19] Kontoyiannis, I. and Meyn, S.P. (2012). Geometric ergodicity and the spectral gap of non-reversible Markov chains. Probab. Theory Related Fields 154 327–339.
  • [20] Liverani, C., Saussol, B. and Vaienti, S. (1999). A probabilistic approach to intermittency. Ergodic Theory Dynam. Systems 19 671–685.
  • [21] Merlevède, F., Peligrad, M. and Utev, S. (2006). Recent advances in invariance principles for stationary sequences. Probab. Surv. 3 1–36.
  • [22] Meyn, S. and Tweedie, R.L. (2009). Markov Chains and Stochastic Stability, 2nd ed. Cambridge: Cambridge Univ. Press.
  • [23] Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics 83. Cambridge: Cambridge Univ. Press.
  • [24] Poznyak, A.S. (2001). The strong law of large numbers for dependent vector processes with decreasing correlation: “double averaging concept”. Math. Probl. Eng. 7 87–95.
  • [25] Revuz, D. (1975). Markov Chains. North-Holland Mathematical Library 11. Amsterdam: North-Holland.
  • [26] Sarig, O. (2002). Subexponential decay of correlations. Invent. Math. 150 629–653.
  • [27] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Berlin: Springer.
  • [28] Yosida, K. and Kakutani, S. (1941). Operator-theoretical treatment of Markoff’s process and mean ergodic theorem. Ann. of Math. (2) 42 188–228.
  • [29] Young, L.-S. (1999). Recurrence times and rates of mixing. Israel J. Math. 110 153–188.
  • [30] Zhang, S. (2000). Existence and application of optimal Markovian coupling with respect to non-negative lower semi-continuous functions. Acta Math. Sin. (Engl. Ser.) 16 261–270.