Periodic regeneration

AbstractWe consider stochastic processes Z = (Zt)[0,∞), on a general state space, having a certain periodic regeneration property: there is an increasing sequence of random times (Sn)∞0 such that the post-Sn process is conditionally independent of S0,…,Sn given (Sn mod 1) and the conditional distribution does not depend on n. Our basic condition is that the distributions Ps(A) = P(Sn+1 − Sn ∈ A|Sn = s), s ∈[0, 1), have a common component that is absolutely continuous w.r.t. Lebesgue measure. Then Z has the following time-homogeneous regeneration property: there exists a discrete aperiodic renewal process T = (Tn)∞0 such that the post-Tn process is independent of T0,…,Tn and its distribution does not depend on n; this yields weak ergodicity. Further, the Markov chain (Sn mod 1)∞0 has an invariant distribution π[0,1) and it holds that Tn+1 − Tn has finite first moment if and only if m = ∫ m(Ps)π[0,1)(ds) < ∞ where m(Ps) is the first moment of Ps; this yields periodic ergodicity. Also, some distributional properties of T0 and Tn+1 − Tn are established leading to improved ergodic regults. Finally, a uniform key periodic renewal theorem is derived