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Representations and limit theorems for extreme value distributions

Published online by Cambridge University Press:  14 July 2016

Peter Hall
Peter Hall*
Affiliation:
University of Melbourne
*
∗Now at The Australian National University, Canberra.
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Abstract

Let Xn1 ≦ Xn2 ≦ ··· ≦ Xnn denote the order statistics from a sample of n independent, identically distributed random variables, and suppose that the variables Xnn, Xn, n–1, ···, when suitably normalized, have a non-trivial limiting joint distribution ξ1, ξ2, ···, as n → ∞. It is well known that the limiting distribution must be one of just three types. We provide a canonical representation of the stochastic process {ξn, n ≧ 1} in terms of exponential variables, and use this representation to obtain limit theorems for ξ n as n →∞.

Keywords

EXTREME VALUE DISTRIBUTIONSEXTREMAL PROCESSESREPRESENTATIONSLIMIT THEOREMS
Type
Short Communications
Information
Journal of Applied Probability , Volume 15 , Issue 3 , September 1978 , pp. 639 - 644
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Barbour, A. D. (1974) Tail sums of convergent series of independent random variables. Proc. Camb. Phil. Soc. 75, 361–364.Google Scholar
[2] Dwass, M. (1966) Extremal processes, II. Illinois J. Math. 10, 381–391.Google Scholar
[3] Gnedenko, B. V. (1943) Sur la distribution limite du terme maximum d'une série aléatoire. Ann. Math. 44, 423–453.Google Scholar
[4] Haan, L. De (1970) On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Mathematical Centre Tracts 32, Amsterdam.Google Scholar
[5] Heyde, C. C. (1977) A contribution to the problem of rates of convergence for the martingale convergence theorem. J. Appl. Prob. 14, 758–775.CrossRefGoogle Scholar
[6] Resnick, S. I. and Rubinovitch, M. (1973) The structure of extremal processes. Adv. Appl. Prob. 5, 287–307.Google Scholar
[7] Renyi, A. (1953) On the theory of order statistics. Acta Math. Acad. Sci. Hung. 4, 191–231.CrossRefGoogle Scholar
[8] Smid, B. and Stam, A. J. (1975) Convergence in distribution of quotients of order statistics. Stoch. Proc. Appl. 3, 287–292.Google Scholar
[9] Smirnov, N. V. (1952) Limit distributions for the terms of a variational series. Amer. Math. Soc. Trans. 67, 82–143.Google Scholar
[10] Whittaker, E. T. and Watson, G. N. (1927) A Course of Modern Analysis , 4th edn. Cambridge University Press, London.Google Scholar