Abstract: We show that certain Foster-type drift conditions related to the existence of a stationary measure for a Markov chain remain valid without any continuity or irreducibility assumptions, provided a uniform countable additivity condition is satisfied. This condition holds, for example, if the transition densities are suitably bounded. Examples show that this condition covers classes of chains not previously addressed. We apply the methods to various non-linear time series models.
Key words and phrases: Bilinear models, ergodicity, Foster-Lyapunov criteria, geometric ergodicity, invariant measures, irreducibility, nonlinear ARMA models, positive recurrence, stationary measures.