Periodic regeneration

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