The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory

We derive sufficient conditions for [is proportional to] [lambda] (dx)||Pn(x, ·) - [pi]|| to be of order o([psi](n)-1), where Pn (x, A) are the transition probabilities of an aperiodic Harris recurrent Markov chain, [pi] is the invariant probability measure, [lambda] an initial distribution and [psi] belongs to a suitable class of non-decreasing sequences. The basic condition involved is the ergodicity of order [psi], which in a countable state space is equivalent to [Sigma] [psi](n)Pi{[tau]i[greater-or-equal, slanted]n} <[infinity] for some i, where [tau]i is the hitting time of the tate i. We also show that for a general Markov chain to be ergodic of order [psi] it suffices that a corresponding condition is satisfied by a small set. We apply these results to non-singular renewal measures on providing a probabilisite method to estimate the right tail of the renewal measure when the increment distribution F satisfies [is proportional to] tF(dt) 0; > 0 and [is proportional to] [psi](t)(1- F(t))dt< [infinity].