Stability of functionals of perturbed Markov chains under the condition of uniform minorization

In this paper, we investigate the stability of functionals and trajectories of two different, independent, time-inhomogeneous, discrete-time Markov chains on a general state space. We obtain various stability estimates such as an estimate for a difference in expectations of functionals, {L_{2}} stability, and a probability of large deviations. The key condition that is used is the minorization condition on the whole space. We consider different limitations on the functional and on the proximity of two chains. We use the coupling method as a primary technique in our proofs.

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

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.

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

Suppose that {X t } 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 {X t } 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.