The coupling of regenerative processes

Abstract

A distributional coupling concept is defined for continuous-time stochastic processes on a general state space and applied to processes having a certain non-time-homogeneous regeneration property: regeneration occurs at random times S o, S 1, · ·· forming an increasing Markov chain, the post-Sn process is conditionally independent of S o, · ··, Sn –1 given Sn , and the conditional distribution is independent of n. The coupling problem is reduced to an investigation of the regeneration times S o, S 1, · ··, and a successful coupling is constructed under the condition that the recurrence times Xn+ 1 = Sn +1 – Sn given that , are stochastically dominated by an integrable random variable, and that the distributions , have a common component which is absolutely continuous with respect to Lebesgue measure (or aperiodic when the Sn 's are lattice-valued). This yields results on the tendency to forget initial conditions as time tends to ∞. In particular, tendency towards equilibrium is obtained, provided the post-Sn process is independent of Sn . The ergodic results cover convergence and uniform convergence of distributions and mean measures in total variation norm. Rate results are also obtained under moment conditions on the Ps 's and the times of the first regeneration.