Let {ξn} be a non-decreasing stochastically monotone Markov
chain whose transition probability Q(.,.) has Q(x,{x})=β(x)>0 for
some function β(.) that is non-decreasing with β(x)↑1 as x→+∞, and
each Q(x,.) 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.