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Two limit theorems for ergodically regenerated stochastic processes

Published online by Cambridge University Press:  24 October 2008

Joseph Kupka
Joseph Kupka
Affiliation:
Monash University, Clayton, Victoria 3168, Australia
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Abstract

There is a subtle difference between the recurrent event of Feller and the regenerative phenomenon of Kingman: The former regenerates an entire ambient process, whereas the latter regenerates only itself. This paper generalizes Feller's definition to a discrete regenerative phenomenon E in association with an arbitrary discrete-time stochastic process X. Two limit theorems of a general character are proved for the process X when it is regenerated by an ergodic phenomenon E. The first implies that X becomes strictly stationary at a uniform rate which is determined solely by the asymptotic behaviour of E. The second is essentially a mixing convergence theorem which implies the asymptotic independence of regenerating and regenerated events. Applications include (1) the outright independence of the regenerating events and the regenerated events which occur ‘at the end of time’, and (2) the identification of the transient features of X with the null sets of the stationary limiting probability. Numerous open questions are posed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 3 , May 1983 , pp. 519 - 536
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Aldous, D. J. and Eagleson, G. K.On mixing and stability of limit theorems. Ann. Prob. 6 (1978), 325–331.CrossRefGoogle Scholar
(2)Breiman, L.Probability (Addison-Wesley, Reading, Mass., and London, 1968).Google Scholar
(3)Chatterji, S. D.A note on the convergence of Banach-space valued martingales. Math. Ann. 153 (1964), 142–149.CrossRefGoogle Scholar
(4)Chatterji, S. D.Martingale convergence and the Radon-Nikodym theorem in Banach spaces. Math. Scand. 22 (1968), 21–41.CrossRefGoogle Scholar
(5)Chung, K. L.Markov chains with stationary transition probabilities. Die Grundlehren der mathematischen Wissenschaften, vol. 104 (Springer-Verlag, Berlin and New York, 1960).CrossRefGoogle Scholar
(6)Cohn, D. L.Measure Theory (Birkhäuser, Boston, 1980).CrossRefGoogle Scholar
(7)Dunford, N. and Schwartz, J. T.Linear operators, I: General theory. Pure and Applied Mathematics, vol. 7 (Interscience, New York, 1958).Google Scholar
(8)Feller, W.An introduction to probability theory and its applications, vol. I, 3rd ed. (Wiley, New York and London, 1968).Google Scholar
(9)Ionescu Tulcea, A. and , C. On the existence of a lifting commuting with the left translations of an arbitrary locally compact group. Proc. 5th Berkeley Sympos., vol. II, part 1 (University of California Press, Berkeley, 1967), pp. 63–97.Google Scholar
(10)Ionescu Tulcea, A. and , C. Topics in the theory of lifting. Ergebnisse der Mathematik, vol. 48 (Springer-Verlag, Berlin and New York, 1969).Google Scholar
(11)Iosifescu, M. and Tautu, P.Stochastic processes and Applications in Biology and Medicine, vol. I (Springer-Verlag, Berlin and New York, 1973).Google Scholar
(12)Johnson, R.Strong liftings commuting with minimal distal flows. Pacific J. Math. 90 (1980), 77–85.CrossRefGoogle Scholar
(13)Karlin, S.On the renewal equation. Pacifiic J. Math. 5 (1955), 229–257.CrossRefGoogle Scholar
(14)Kingman, J. F. C.A continuous time analogue of the theory of recurrent events. Bull. Amer. Math. Soc. 69 (1963), 268–272.CrossRefGoogle Scholar
(15)Kingman, J. F. C.The stochastic theory of regenerative events. Zeit. Wahr. verw. Geb. 2 (1964), 180–224.CrossRefGoogle Scholar
(16)Kingman, J. F. C.Regenerative phenomena (Wiley, New York and London, 1972).Google Scholar
(17)Loève, M.Probability Theory, 3rd ed. (D. Van Nostrand, Princeton and London, 1963).Google Scholar
(18)Orey, S.Strong ratio limit property. Bull. Amer. Math. Soc. 67 (1961), 571–574.CrossRefGoogle Scholar
(19)Von Neumann, J. and Stone, M.The determination of representative elements in the residual classes of a Boolean algebra. Fund. Math. 25 (1935), 353–376.CrossRefGoogle Scholar
(20)Walters, P. An introduction to ergodic theory. Graduate Texts in Mathematics, vol. 79 (Springer-Verlag, Berlin and New York, 1982).Google Scholar