The Annals of Applied Probability

A complete solution to Blackwell’s unique ergodicity problem for hidden Markov chains

Pavel Chigansky and Ramon van Handel

Full-text: Open access

Abstract

We develop necessary and sufficient conditions for uniqueness of the invariant measure of the filtering process associated to an ergodic hidden Markov model in a finite or countable state space. These results provide a complete solution to a problem posed by Blackwell (1957), and subsume earlier partial results due to Kaijser, Kochman and Reeds. The proofs of our main results are based on the stability theory of nonlinear filters.

Article information

Source
Ann. Appl. Probab. Volume 20, Number 6 (2010), 2318-2345.

Dates
First available in Project Euclid: 19 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1287494562

Digital Object Identifier
doi:10.1214/10-AAP688

Mathematical Reviews number (MathSciNet)
MR2759736

Zentralblatt MATH identifier
1202.93159

Subjects
Primary: 93E11: Filtering [See also 60G35]
Secondary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 60J05: Discrete-time Markov processes on general state spaces 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 93E15: Stochastic stability

Keywords
Hidden Markov models filtering unique ergodicity asymptotic stability

Citation

Chigansky, Pavel; van Handel, Ramon. A complete solution to Blackwell’s unique ergodicity problem for hidden Markov chains. Ann. Appl. Probab. 20 (2010), no. 6, 2318--2345. doi:10.1214/10-AAP688. https://projecteuclid.org/euclid.aoap/1287494562


Export citation

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] Blackwell, D. and Dubins, L. (1962). Merging of opinions with increasing information. Ann. Math. Statist. 33 882–886.
  • [4] Budhiraja, A. (2003). Asymptotic stability, ergodicity and other asymptotic properties of the nonlinear filter. Ann. Inst. H. Poincaré Probab. Statist. 39 919–941.
  • [5] Chigansky, P., Liptser, R. and van Handel, R. (2010). Intrinsic methods in filter stability. In The Oxford University Handbook of Nonlinear Filtering (D. Crisan and B. Rozovsky, eds.). Oxford Univ. Press. To appear.
  • [6] Di Masi, G. B. and Stettner, Ł. (2005). Ergodicity of hidden Markov models. Math. Control Signals Systems 17 269–296.
  • [7] Jakubowski, A. (1988). Tightness criteria for random measures with application to the principle of conditioning in Hilbert spaces. Probab. Math. Statist. 9 95–114.
  • [8] Kaijser, T. (1975). A limit theorem for partially observed Markov chains. Ann. Probab. 3 677–696.
  • [9] Kaijser, T. (2009). On Markov chains induced by partitioned transition probability matrices. Preprint. Available at http://arxiv.org/abs/0907.4502.
  • [10] Kochman, F. and Reeds, J. (2006). A simple proof of Kaijser’s unique ergodicity result for hidden Markov α-chains. Ann. Appl. Probab. 16 1805–1815.
  • [11] Kunita, H. (1971). Asymptotic behavior of the nonlinear filtering errors of Markov processes. J. Multivariate Anal. 1 365–393.
  • [12] Stettner, Ł. (1989). On invariant measures of filtering processes. In Stochastic Differential Systems (Bad Honnef, 1988). Lecture Notes in Control and Inform. Sci. 126 279–292. Springer, Berlin.
  • [13] van Handel, R. (2009). Observability and nonlinear filtering. Probab. Theory Related Fields 145 35–74.
  • [14] van Handel, R. (2009). The stability of conditional Markov processes and Markov chains in random environments. Ann. Probab. 37 1876–1925.
  • [15] van Handel, R. (2009). Uniform observability of hidden Markov models and filter stability for unstable signals. Ann. Appl. Probab. 19 1172–1199.