## Ergodic behaviour of nonstationary regenerative processes

HTML articles powered by AMS MathViewer

- by David McDonald
- Trans. Amer. Math. Soc.
**255**(1979), 135-152 - DOI: https://doi.org/10.1090/S0002-9947-1979-0542874-3
- PDF | Request permission

## 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

*n*th generation and ${\alpha _n}$ be the mean time ${V_t}$ is in

*A*during the

*n*th 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 $\lim \limits _{n \to \infty } (1/n)\Sigma _{j = 1}^n {\alpha _j} = \alpha$.

## References

- David R. McDonald,
*On local limit theorem for integer valued random variables*, Teor. Veroyatnost. i Primenen.**24**(1979), no. 3, 607–614 (English, with Russian summary). MR**541375** - David McDonald,
*On semi-Markov and semi-regenerative processes. I*, Z. Wahrsch. Verw. Gebiete**42**(1978), no. 4, 261–277. MR**491492**, DOI 10.1007/BF00533463 - David McDonald,
*On semi-Markov and semi-regenerative processes. I*, Z. Wahrsch. Verw. Gebiete**42**(1978), no. 4, 261–277. MR**491492**, DOI 10.1007/BF00533463 - J. Mineka,
*A criterion for tail events for sums of independent random variables*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**25**(1972/73), 163–170. MR**350890**, DOI 10.1007/BF00535889 - J. Mineka,
*Local limit theorems and recurrence conditions for sums of independent integer-valued random variables*, Ann. Math. Statist.**43**(1972), 251–259. MR**314096**, DOI 10.1214/aoms/1177692718 - A. B. Muhin,
*The local limit theorem for densities, and asymptotic uniform distribedness*, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk**15**(1971), no. 1, 17–23 (Russian, with Uzbek summary). MR**0290434** - Steven Orey,
*Tail events for sums of independent random variables*, J. Math. Mech.**15**(1966), 937–951. MR**0202178** - Walter L. Smith,
*Renewal theory and its ramifications*, J. Roy. Statist. Soc. Ser. B**20**(1958), 243–302. MR**99090**, DOI 10.1111/j.2517-6161.1958.tb00294.x

## Bibliographic Information

- © Copyright 1979 American Mathematical Society
- 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