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Limit laws for kth order statistics from strong-mixing processes

Published online by Cambridge University Press:  14 July 2016

W. Dziubdziela
W. Dziubdziela*
Affiliation:
University of Wrocław
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Abstract

We present necessary and sufficient conditions for the weak convergence of the distributions of the kth order statistics from a strictly stationary strong-mixing sequence of random variables to limit laws which are represented in terms of a compound Poisson distribution. The obtained limit laws form a class larger than that occurring in the independent case.

Keywords

WEAK CONVERGENCECOMPOUND POISSON DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 4 , December 1984 , pp. 720 - 729
Copyright
Copyright © Applied Probability Trust 1984 

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References

Adler, R. J. (1978) Weak convergence results for extremal processes generated by dependent random variables. Ann. Prob. 6, 660–667.Google Scholar
Chernick, M. R. (1981) A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Ann. Prob. 9, 145–149.Google Scholar
Davis, R. A. (1982) Limit laws for the maximum and minimum of stationary sequences. Z. Wahrscheinlichkeitsth. 61, 31–42.Google Scholar
Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
Haiman, M. G. (1981) Valeurs extrémales de suites stationnaires de variables aléatoires m-dépendantes. Ann. Inst. Henri Poincaré 17, 309–330.Google Scholar
Ibragimov, I. A. and Linnik, Yu. V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhofl, Groningen.Google Scholar
Leadbetter, M. R., Lindgren, G. and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
Loynes, R. M. (1965) Extreme values in uniformly mixing stationary stochastic processes. Ann. Math. Statist. 36, 993–999.Google Scholar
Rényi, A. (1951) On composed Poisson distribution II. Acta Math. Acad. Sci. Hung. 2, 83–98.Google Scholar
Welsch, R. E. (1972) Limit laws for extreme order statistics from strong-mixing processes. Ann. Math. Statist. 43, 439–446.Google Scholar