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

Daren B.H. Cline

Abstract   Suppose 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 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 often can be embedded into Feller chains, allowing us to apply the conclusions above.

This paper currently is under review.  Please click on my name to email a request for a preprint.