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Journal of Applied Mathematics and Simulation
Volume 1 (1987), Issue 2, Pages 137-154
http://dx.doi.org/10.1155/S1048953388000103

Remarks on a monotone Markov chain

Statistics and Applied Probability Program, University of California, Santa Barbara, USA

Received 27 November 1987; Accepted 1 November 1988

Copyright © 1987 Hindawi Publishing Corporation. 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

In applications, considerations on stochastic models often involve a Markov chain {ζn}0 with state space in R+, and a transition probability Q. For each x  R+ the support of Q(x,.) is [0,x]. This implies that ζ0ζ1. Under certain regularity assumptions on Q we show that Qn(x,Bu)1 as n for all u>0 and that 1Qn(x,Bu)[1Q(x,Bu)]n where Bu=[0,u). Set τ0=max{k;ζk=ζ0}, τn=max{k;ζk=ζτn1+1} and write Xn=ζτn1+1, Tn=τnτn1. We investigate some properties of the imbedded Markov chain {Xn}0 and of {Tn}0. We determine all the marginal distributions of {Tn}0 and show that it is asymptotically stationary and that it possesses a monotonicity property. We also prove that under some mild regularity assumptions on β(x)=1Q(x,Bx), 1n(Tia)/bndZN(0,1).