Ergodic behaviour of nonstationary regenerative processes

Abstract:

Let ${V_t}$ be a regenerative process whose successive generations are not necessarily identically distributed and let A be a measurable set in the range of ${V_t}$. Let ${\mu _n}$ be the mean length of the nth generation and ${\alpha _n}$ be the mean time ${V_t}$ is in A during the nth generation. We give conditions ensuring ${\lim _{t \to \infty }} \operatorname {prob} \{ {V_t} \in A \} = \alpha /\mu$ where $\lim \limits _{n \to \infty } (1/n)\Sigma _{j = 1}^n {\mu _j} = \mu$ and $\lim \limits _{n \to \infty } (1/n)\Sigma _{j = 1}^n {\alpha _j} = \alpha$.