Letbe a non-decreasing Markov chain with a transition probability Q satisfying.Assuming that Q(x,By),where,is non-increasing in x, and that Q(x,By+x) B is non-decreasing in x,we prove that,where {Zi} is stochastically non-increasing and asymptotically i.i.d.,possessing a mixin property.We use this to show that(a.s) and thatin probability,if,where T(y) is the hitting time of (y,∞).We also show thatis Markov renewal process whereare sojourn times of