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On strong mixing and Leadbetter's D condition

Published online by Cambridge University Press:  14 July 2016

Michael R. Chernick
Michael R. Chernick*
Affiliation:
The Aerospace Corporation
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Abstract

Strong mixing is a condition which is often assumed to prove limit theorems for strictly stationary processes. Leadbetter's condition D(un) is used to prove limit theorems for maxima of stationary processes.

A sufficient condition for strong mixing to hold is given for the case where the process satisfies a pth-order Markov property. This condition can be easy to check for when p is small. This point is illustrated by two examples of first-order autoregressive processes.

The condition D(un) is shown to hold for any stationary Markov process.

Keywords

MARKOV PROCESSESPTH-ORDER MARKOV PROPERTYAUTOREGRESSIVE PROCESSESSTRONG MIXING CONDITIONDISTRIBUTIONAL MIXING CONDITIONASYMPTOTIC DISTRIBUTION THEORYSTRICTLY STATIONARY PROCESSES
Type
Short Communications
Information
Journal of Applied Probability , Volume 18 , Issue 3 , September 1981 , pp. 764 - 769
Copyright
Copyright © Applied Probability Trust 

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References

Campbell, K. (1979) Recursive computation of M-estimates for the parameters of a finite autoregression. Unpublished paper, Los Alamos Scientific Laboratory.Google Scholar
Chernick, M. R. (1977) A limit theorem for the maximum of an exponential autoregressive process. SIMS Technical Report No. 14, Stanford University.Google Scholar
Gastwirth, J. L. and Rubin, H. (1975) The asymptotic distribution theory of the empiric CDF for mixing stochastic processes. Ann. Statist. 3, 809–824.CrossRefGoogle Scholar
Jacobs, P. A. and Lewis, P. A. W. (1977) A mixed autoregressive moving average exponential sequence and point process (EARMA 1,1). Adv. Appl. Prob. 9, 87–104.Google Scholar
Leadbetter, M. R. (1974) On extreme values in stationary sequences. Z. Wahrscheinlichkeitsth. 28, 289–303.CrossRefGoogle Scholar
Loynes, R. M. (1965) Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999.CrossRefGoogle Scholar
Rosenblatt, M. (1956) A central limit theorem and a strong mixing condition. Proc. Nat. Acad. Sci. USA 42, 43–47.Google Scholar
Royden, H. (1968) Real Analysis, 2nd edn. Macmillan Company, New York.Google Scholar