In this brief paper we find computable exponential convergence rates for a large class of stochastically ordered Markov processes. We extend the result of Lund, Meyn, and Tweedie (1996), who found exponential convergence rates for stochastically ordered Markov processes starting from a fixed initial state, by allowing for a random initial condition that is also stochastically ordered. Our bounds are formulated in terms of moment-generating functions of hitting times. To illustrate our result, we find an explicit exponential convergence rate for an M M 1 queue beginning in equilibrium and then experiencing a change in its arrival or departure rates, a setting which has not been studied to our knowledge.
Other works build on the derivation of more general convergence bounds. For example, (2007) applied the main theorem in order to get bounds on the best uniform convergence rate for strongly ergodic Markov chains. Liu2007 Drawing on the stochastic monotonicity and coupling approach of (1996), Roberts and Rosenthal (2017) derived convergence rate bounds for symmetric Langevin diffusions, while Sarantsev (2016) followed the Lyapunov function approach to find convergence rates for jump diffusions on the half-line. Roberts2017, Sarantsev2016 (2005) also used a coupling method, focusing on establishing subgeometric convergence rates. Hou2005 In related work to @cite_1 , (2010) established subgeometric convergence rates via first hitting times and drift functions. Liu2010 (2004) were able to generalize convergence bounds to time-inhomogeneous chains using coupling and drift conditions. Douc2004
@cite_1
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Let (Phi(t))(t is an element of R+) be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure pi. We investigate the rates of convergence of the transition function P-t(x, (.)) to pi; specifically, we find conditions under which r(t) vertical bar vertical bar P-t (x, (.)) - pi vertical bar vertical bar -> 0 as t -> infinity, for suitable subgeometric rate functions r(t), where vertical bar vertical bar - vertical bar vertical bar denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.