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