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Geometric convergence rates for stochastically ordered Markov chains. (English) Zbl 0847.60053
Summary: Let \(\{\Phi_n\}\) be a Markov chain on the state space \([0, \infty)\) 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 \(\{\Phi_n \}\) in total variation; that is, proving the existence of a limiting probability measure \(\pi\) and a number \(r > 1\) such that \[ \lim_{n \to \infty} r^n \sup_{A \in {\mathcal B} [0, \infty)} \bigl |P_x [\Phi_n \in A] - \pi (A) \bigr |= 0 \] for every deterministic initial state \(\Phi_0 \equiv x\). 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:
60J25 Continuous-time Markov processes on general state spaces
60K25 Queueing theory (aspects of probability theory)
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