Remote Access Transactions of the American Mathematical Society
Green Open Access

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
MathSciNet review: 542874
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $ \mathop {\lim }\limits_{n \to \infty } (1/n)\Sigma _{j = 1}^n\,{\mu _j}\, = \mu $ and $ \mathop {\lim }\limits_{n \to \infty } (1/n)\Sigma _{j = 1}^n\,{\alpha _j}\, = \,\alpha $.


References [Enhancements On Off] (What's this?)

  • [1] 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
  • [2] David McDonald, On semi-Markov and semi-regenerative processes. I, Z. Wahrsch. Verw. Gebiete 42 (1978), no. 4, 261–277. MR 491492, 10.1007/BF00533463
  • [3] David McDonald, On semi-Markov and semiregenerative processes. II, Ann. Probab. 6 (1978), no. 6, 995–1014 (1979). MR 512416
  • [4] J. Mineka, A criterion for tail events for sums of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 163–170. MR 0350890
  • [5] J. Mineka, Local limit theorems and recurrence conditions for sums of independent integer-valued random variables, Ann. Math. Statist. 43 (1972), 251–259. MR 0314096
  • [6] 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
  • [7] Steven Orey, Tail events for sums of independent random variables, J. Math. Mech. 15 (1966), 937–951. MR 0202178
  • [8] Walter L. Smith, Renewal theory and its ramifications, J. Roy. Statist. Soc. Ser. B 20 (1958), 243–302. MR 0099090

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60K05

Retrieve articles in all journals with MSC: 60K05


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0542874-3
Keywords: Nonstationary regenerative limits
Article copyright: © Copyright 1979 American Mathematical Society