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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Ergodic behaviour of nonstationary regenerative processes


Author: David McDonald
Journal: Trans. Amer. Math. Soc. 255 (1979), 135-152
MSC: Primary 60K05
DOI: https://doi.org/10.1090/S0002-9947-1979-0542874-3
MathSciNet review: 542874
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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 $\lim \limits _{n \to \infty } (1/n)\Sigma _{j = 1}^n {\alpha _j} = \alpha$.


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Keywords: Nonstationary regenerative limits
Article copyright: © Copyright 1979 American Mathematical Society