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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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