Let {ξn} be a non-decreasing stochastically monotone Markov chain whose transition probability Q(ë,ë) has Q(x,{x})=β(x)>0 for some function β(ë) that is non-decreasing with β(x)ℝ1 as x⇒¸+•, and each Q(x,ë) is non-atomic otherwise. A typical realization of {ξn} is a Markov renewal process {(Xn,Tn)}, where ξj=Xn for Tn consecutive values of j,Tn geometric on {1,2,...} with parameter β(Xn). Conditions are given for Xn to be relatively stable and for Tn to be weakly convergent.
