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On the second Borel-Cantelli lemma for strongly mixing sequences of events

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Dirk Tasche
Dirk Tasche*
Affiliation:
Technische Universität Berlin
*
∗Postal address: Ingeborgstrasse 62, 81825 München, Germany.
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Abstract

Assume a given sequence of events to be strongly mixing at a polynomial or exponential rate. We show that the conclusion of the second Borel-Cantelli lemma holds if the series of the probabilities of the events diverges at a certain rate depending on the mixing rate of the events. An application to necessary moment conditions for the strong law of large numbers is given.

Keywords

BOREL-CANTELLI LEMMASTRONG MIXINGSTRONG LAW OF LARGE NUMBERS

MSC classification

Primary: 60F20: Zero-one laws
Secondary: 60F15: Strong theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 381 - 394
Copyright
Copyright © Applied Probability Trust 1997 

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References

Blum, J. R., Hanson, D. L. and Koopmans, L. H. (1963) On the strong law of large numbers for a class of stochastic processes. Z. Wahrscheinlichkeitsth. 2, 1–11.CrossRefGoogle Scholar
Durrett, R. (1991) Probability: Theory and Examples. Wadsworth, Belmont, CA.Google Scholar
Hall, P. and Heyde, C. C. (1980) Martingale Limit Theory and its Application. Academic Press, Boston.Google Scholar
Iosifescu, M. and Theodorescu, R. (1969) Random Processes and Learning. Springer, Berlin.Google Scholar
Kesten, H. and O'Brien, G. L. (1976) Examples of mixing sequences. Duke Math. J. 43, 405–415.CrossRefGoogle Scholar
Rieders, E. (1993) The size of the averages of strongly mixing random variables. Statist. Prob. Lett. 18, 57–64.Google Scholar
Rio, E. (1993) Covariance inequalities for strongly mixing processes. Ann. Inst. H. Poincaré 29, 587–597.Google Scholar
Rio, E. (1995) A maximal inequality and dependent Marcinkiewicz-Zygmund strong laws. Ann. Prob. 23, 918–937.CrossRefGoogle Scholar
Tasche, D. (1995) First-passage time moments of strongly mixing stationary sequences. Preprint 452. Technical University of Berlin.Google Scholar
Yoshihara, K. (1979) The Borel-Cantelli lemma for strong mixing sequences of events and their applications to LIL. Kodai Math. J. 2, 148–157.CrossRefGoogle Scholar