The Annals of Applied Probability

Ergodicity and stability of the conditional distributions of nondegenerate Markov chains

Xin Thomson Tong and Ramon van Handel

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Abstract

We consider a bivariate stationary Markov chain $(X_{n},Y_{n})_{n\ge0}$ in a Polish state space, where only the process $(Y_{n})_{n\ge0}$ is presumed to be observable. The goal of this paper is to investigate the ergodic theory and stability properties of the measure-valued process $(\Pi_{n})_{n\ge0}$, where $\Pi_{n}$ is the conditional distribution of $X_{n}$ given $Y_{0},\ldots,Y_{n}$. We show that the ergodic and stability properties of $(\Pi_{n})_{n\ge0}$ are inherited from the ergodicity of the unobserved process $(X_{n})_{n\ge0}$ provided that the Markov chain $(X_{n},Y_{n})_{n\ge0}$ is nondegenerate, that is, its transition kernel is equivalent to the product of independent transition kernels. Our main results generalize, subsume and in some cases correct previous results on the ergodic theory of nonlinear filters.

Article information

Source
Ann. Appl. Probab. Volume 22, Number 4 (2012), 1495-1540.

Dates
First available in Project Euclid: 10 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1344614202

Digital Object Identifier
doi:10.1214/11-AAP800

Mathematical Reviews number (MathSciNet)
MR2985168

Zentralblatt MATH identifier
1252.60069

Subjects
Primary: 60J05: Discrete-time Markov processes on general state spaces 28D99: None of the above, but in this section
Secondary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 93E11: Filtering [See also 60G35] 93E15: Stochastic stability

Keywords
Nonlinear filtering unique ergodicity asymptotic stability nondegenerate Markov chains exchange of intersection and supremum Markov chain in random environment

Citation

Tong, Xin Thomson; van Handel, Ramon. Ergodicity and stability of the conditional distributions of nondegenerate Markov chains. Ann. Appl. Probab. 22 (2012), no. 4, 1495--1540. doi:10.1214/11-AAP800. https://projecteuclid.org/euclid.aoap/1344614202


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References

  • [1] Baxendale, P., Chigansky, P. and Liptser, R. (2004). Asymptotic stability of the Wonham filter: Ergodic and nonergodic signals. SIAM J. Control Optim. 43 643–669 (electronic).
  • [2] Blackwell, D. (1957). The entropy of functions of finite-state Markov chains. In Transactions of the First Prague Conference on Information Theory, Statistical Decision Functions, Random Processes Held at Liblice Near Prague from November 28 to 30, 1956 13–20. Publishing House of the Czechoslovak Academy of Sciences, Prague.
  • [3] Budhiraja, A. (2002). On invariant measures of discrete time filters in the correlated signal-noise case. Ann. Appl. Probab. 12 1096–1113.
  • [4] Budhiraja, A. (2003). Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter. Ann. Inst. Henri Poincaré Probab. Stat. 39 919–941.
  • [5] Budhiraja, A. and Kushner, H. J. (2001). Monte Carlo algorithms and asymptotic problems in nonlinear filtering. In Stochastics in Finite and Infinite Dimensions 59–87. Birkhäuser, Boston, MA.
  • [6] Cappé, O., Moulines, E. and Rydén, T. (2005). Inference in Hidden Markov Models. Springer, New York.
  • [7] Chaumont, L. and Yor, M. (2003). Exercises in Probability. Cambridge Series in Statistical and Probabilistic Mathematics 13. Cambridge Univ. Press, Cambridge.
  • [8] Crisan, D. and Rozovsky, B. (2011). The Oxford Handbook of Nonlinear Filtering. Oxford Univ. Press, Oxford.
  • [9] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B. Theory of Martingales. North-Holland Mathematics Studies 72. North-Holland, Amsterdam.
  • [10] Di Masi, G. B. and Stettner, Ł. (2005). Ergodicity of hidden Markov models. Math. Control Signals Systems 17 269–296.
  • [11] Douc, R., Moulines, É. and Rydén, T. (2004). Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 2254–2304.
  • [12] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74. Cambridge Univ. Press, Cambridge.
  • [13] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Springer, New York.
  • [14] Kunita, H. (1971). Asymptotic behavior of the nonlinear filtering errors of Markov processes. J. Multivariate Anal. 1 365–393.
  • [15] Kunita, H. (1991). Ergodic properties of nonlinear filtering processes. In Spatial Stochastic Processes. Progress in Probability 19 233–256. Birkhäuser, Boston, MA.
  • [16] Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [17] Ocone, D. and Pardoux, E. (1996). Asymptotic stability of the optimal filter with respect to its initial condition. SIAM J. Control Optim. 34 226–243.
  • [18] Orstein, D. and Sucheston, L. (1970). An operator theorem on $L_{1}$ convergence to zero with applications to Markov kernels. Ann. Math. Statist. 41 1631–1639.
  • [19] Stettner, Ł. (1989). On invariant measures of filtering processes. In Stochastic Differential Systems (Bad Honnef, 1988). Lecture Notes in Control and Information Sciences 126 279–292. Springer, Berlin.
  • [20] van Handel, R. (2009). The stability of conditional Markov processes and Markov chains in random environments. Ann. Probab. 37 1876–1925.
  • [21] van Handel, R. (2009). Uniform time average consistency of Monte Carlo particle filters. Stochastic Process. Appl. 119 3835–3861.
  • [22] van Handel, R. (2012). On the exchange of intersection and supremum of $\sigma$-fields in filtering theory. Israel J. Math. To appear.
  • [23] Volkonskiĭ, V. A. and Rozanov, Y. A. (1959). Some limit theorems for random functions. I. Theory Probab. Appl. 4 178–197.
  • [24] von Weizsäcker, H. (1983). Exchanging the order of taking suprema and countable intersections of $\sigma $-algebras. Ann. Inst. H. Poincaré Sect. B (N.S.) 19 91–100.
  • [25] Yor, M. (1977). Sur les théories du filtrage et de la prédiction. In Séminaire de Probabilités, XI (Univ. Strasbourg, Strasbourg, 1975/1976). Lecture Notes in Math. 581 257–297. Springer, Berlin.