Bounds on regeneration times and convergence rates for Markov chains

In many applications of Markov chains, and especially in Markov chain Monte Carlo algorithms, the rate of convergence of the chain is of critical importance. Most techniques to establish such rates require bounds on the distribution of the random regeneration time T that can be constructed, via splitting techniques, at times of return to a “small set” C satisfying a minorisation condition P(x,·)⩾εϕ(·), x∈C. Typically, however, it is much easier to get bounds on the time τC of return to the small set itself, usually based on a geometric drift function PV⩽λV+b1C, where PV(x)=Ex(V(X1)). We develop a new relationship between T and τC, and this gives a bound on the tail of T, based on ε,λ and b, which is a strict improvement on existing results. When evaluating rates of convergence we see that our bound usually gives considerable numerical improvement on previous expressions.