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Irreducibility and continuity assumptions for positive operators with application to threshold GARCH time series models

Part of: Markov processes Mathematical economics Ergodic theory Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Daren B. H. Cline
Daren B. H. Cline*
Affiliation:
Texas A&M University
*
∗ Postal address: Department of Statistics, Texas A&M University, College Station TX 77843-3143, USA. Email address: dcline@stat.tamu.edu
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Abstract

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Suppose that {Xt} is a Markov chain such as the state space model for a threshold GARCH time series. The regularity assumptions for a drift condition approach to establishing the ergodicity of {Xt} typically are ϕ-irreducibility, aperiodicity, and a minorization condition for compact sets. These can be very tedious to verify due to the discontinuous and singular nature of the Markov transition probabilities. We first demonstrate that, for Feller chains, the problem can at least be simplified to focusing on whether the process can reach some neighborhood that satisfies the minorization condition. The results are valid not just for the transition kernels of Markov chains but also for bounded positive kernels, opening the possibility for new ergodic results. More significantly, we show that threshold GARCH time series and related models of interest can often be embedded into Feller chains, allowing us to apply the conclusions above.

Keywords

Feller operatorT-chainGARCHthreshold time series

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 37A50: Relations with probability theory and stochastic processes 62M10: Time series, auto-correlation, regression, etc. 91B84: Economic time series analysis
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 49 - 76
Copyright
Copyright © Applied Probability Trust 2011 

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