Journal of Applied Mathematics and Stochastic Analysis
Volume 4 (1991), Issue 4, Pages 293-303
doi:10.1155/S1048953391000229
Relative stability and weak convergence in non-decreasing
stochastically monotone Markov chains
University of California, Department of Statistics and Applied Probability, Santa Barbara, CA, USA
Received 1 January 1991; Revised 1 June 1991
Copyright © 1991 P. Todorovic. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
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.