Journal of Applied Mathematics and Stochastic Analysis
Volume 4 (1991), Issue 4, Pages 293-303

Relative stability and weak convergence in non-decreasing stochastically monotone Markov chains

P. Todorovic

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.


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.