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Geometric convergence rates for stochastically ordered Markov chains. (English) Zbl 0847.60053

Summary: Let {Φ n } be a Markov chain on the state space [0,) that is stochastically ordered in its initial state; that is, a stochastically larger initial state produces a stochastically larger chain at all other times. Examples of such chains include random walks, the number of customers in various queueing systems, and a plethora of storage processes. A large body of recent literature concentrates on establishing geometric ergodicity of {Φ n } in total variation; that is, proving the existence of a limiting probability measure π and a number r>1 such that

lim n r n sup A[0,) P x [Φ n A] - π (A)=0

for every deterministic initial state Φ 0 x. We seek to identify the largest r that satisfies this relationship. A dependent sample path coupling and a Foster-Lyapunov drift inequality are used to derive convergence rate bounds; we then show that the bounds obtained are frequently the best possible. Application of the methods to queues and random walks are included.

MSC:
60J25Continuous-time Markov processes on general state spaces
60K25Queueing theory