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March 2011 Irreducibility and continuity assumptions for positive operators with application to threshold GARCH time series models
Daren B. H. Cline
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Adv. in Appl. Probab. 43(1): 49-76 (March 2011). DOI: 10.1239/aap/1300198512

Abstract

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.

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Daren B. H. Cline. "Irreducibility and continuity assumptions for positive operators with application to threshold GARCH time series models." Adv. in Appl. Probab. 43 (1) 49 - 76, March 2011. https://doi.org/10.1239/aap/1300198512

Information

Published: March 2011
First available in Project Euclid: 15 March 2011

zbMATH: 1238.62101
MathSciNet: MR2761147
Digital Object Identifier: 10.1239/aap/1300198512

Subjects:
Primary: 60J05
Secondary: 37A50 , 62M10 , 91B84

Keywords: Feller operator , GARCH , T-chain , threshold time series

Rights: Copyright © 2011 Applied Probability Trust

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Vol.43 • No. 1 • March 2011
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