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. The paper extends the analysis to show that the distance between the nth-step transition probability and the invariant probability measure is bounded above by ρⁿ(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.