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

P. Todorovic

 University of California, Department of Statistics and Applied Probability, Santa Barbara, CA, USA
Received 1 January 1991; Revised 1 June 1991

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.