A note on the geometric ergodicity of a Markov chain

It is known that if an irreducible and aperiodic Markov chain satisfies a ‘drift' condition in terms of a non-negative measurable function g(x), it is geometrically ergodic. See, e.g. Nummelin (1984), p. 90. We extend the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρ n(a + bg(x)) for some constants a, b> 0 and ρ < 1. The result is then applied to obtain convergence rates to the invariant probability measures for an autoregressive process and a random walk on a half line.