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.