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On the uniform ergodicity of Markov processes of order 2

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Ulrich Herkenrath
Ulrich Herkenrath*
Affiliation:
Gerhard-Mercator-Universität Duisburg
*
∗ Postal address: Fakultät 4, Institut für Mathematik, Gerhard-Mercator-Universität Duisburg, Lotharstr. 65, D-47048 Duisburg, Germany. Email address: herkenrath@math.uni-duisburg.de
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Abstract

We study the uniform ergodicity of Markov processes (Zn, n ≥ 1) of order 2 with a general state space (Z, 𝒵). Markov processes of order higher than 1 were defined in the literature long ago, but scarcely treated in detail. We take as the basis for our considerations the natural transition probability Q of such a process. A Markov process of order 2 is transformed into one of order 1 by combining two consecutive variables Z2n–1 and Z2n into one variable Yn with values in the Cartesian product space (Z × Z, 𝒵 ⊗ 𝒵). Thus, a Markov process (Yn, n ≥ 1) of order 1 with transition probability R is generated. Uniform ergodicity for the process (Zn, n ≥ 1) is defined in terms of the same property for (Yn, n ≥ 1). We give some conditions on the transition probability Q which transfer to R and thus ensure the uniform ergodicity of (Zn, n ≥ 1). We apply the general results to study the uniform ergodicity of Markov processes of order 2 which arise in some nonlinear time series models and as sequences of smoothed values in sequential smoothing procedures of Markovian observations. As for the time series models, Markovian noise sequences are covered.

Keywords

Markov process of order 2uniform ergodicitynonlinear time seriesexponential smoothingMarkovian noise sequenceMarkovian innovations

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 2 , June 2003 , pp. 455 - 472
Copyright
Copyright © Applied Probability Trust 2003 

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References

Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Borkovec, M. and Klüppelberg, C. (2001). The tail of the stationary distribution of an autoregressive process with ARCH(1)-errors. Ann. Appl. Prob. 11, 1220–1241.Google Scholar
Bowerman, B. L., and O'Connell, R. T. (1993). Forecasting and Time Series: An Applied Approach. Duxbury, Belmont, CA.Google Scholar
Brown, R. G. (1963). Smoothing, Forecasting and Prediction of Discrete Time Series. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
Herkenrath, U. (1977). Markov processes under continuity assumptions. Rev. Roumaine Math. Pures Appl. 22, 1419–1431.Google Scholar
Herkenrath, U. (1994a). Generalized adaptive exponential smoothing procedures. J. Appl. Prob. 31, 673–690.Google Scholar
Herkenrath, U. (1994b). Premium adjustment by generalized adaptive exponential smoothing. Insurance Math. Econom. 15, 203–217.Google Scholar
Herkenrath, U. (1999). Generalized adaptive exponential smoothing of observations from an ergodic hidden Markov model. J. Appl. Prob. 36, 987–998.CrossRefGoogle Scholar
Iosifescu, M., and Grigorescu, S. (1990). Dependence with Complete Connections and Its Applications. Cambridge University Press.Google Scholar
Iosifescu, M., and Theodorescu, R. (1969). Random Processes and Learning. Springer, Berlin.Google Scholar
Meyn, S. P., and Tweedie, R. L. (1996). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Royden, H. L. (1968). Real Analysis, 2nd edn. Macmillan, New York.Google Scholar
Tong, H. (1995). Non-Linear Time Series. Oxford University Press.Google Scholar
Weiss, A. A. (1984). ARMA models with ARCH errors. J. Time Ser. Anal. 5, 129–143.CrossRefGoogle Scholar