Statistica Sinica 8(1998), 429-443

A RENEWAL THEORY FOR PERTURBED MARKOV

RANDOM WALKS WITH APPLICATIONS TO

SEQUENTIAL ANALYSIS

Yeong-Tzay Su

Kaohsiung Normal University

Abstract: Let Zn be a perturbed Markov random walk. We prove that under certain conditions ΣnP{a<Zn≤a+h} converges to a finite limit, as a→∞, for each h>0. We also present an important class of processes satisfying these conditions and then apply these results to sequential analysis and obtain an expression for the asymptotic value of the expected sample size for a repeated likelihood ratio test problem.

Key words and phrases: Blackwell-type renewal theorem, convergence rate, expected sample size, perturbed Markov random walks, repeated likelihood ratio test.