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On the relative stability of large order statistics

Published online by Cambridge University Press:  24 October 2008

Peter Hall
Peter Hall
Affiliation:
The Australian National University, Canberra
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Abstract

Let X1, X2, … be independent and identically distributed (i.i.d.) random variables and let Xn, r denote the rth largest of X1, X2, …, Xn (so that Xn, r is the (n − r + l)th-order statistic of X1, X2, …, Xn). It is well known that if Xn, l/cn→1 in probability or with probability 1, for some sequence of constants cn, then Xn, r/cn→1 for each r ≥ 1. Therefore if r(n) → ∞ sufficiently slowly, Xn, r(n)/cn→1 for the same sequence of constants cn. In this paper we study behaviour of this type in considerable detail. We find necessary and sufficient conditions on the rate of increase of r(n), n ≤ 1, for the limit theorem Xn, r(n)/cn→1 to hold, and we investigate the rate of convergence in terms of a central limit theorem and a law of the iterated logarithm (LIL). The LIL takes a particularly interesting form, and there are five distinctly different modes of behaviour.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 3 , November 1979 , pp. 467 - 475
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCES

(0)Balkema, A. A. and De Haan, L.Limit theorems for order statistics. II. Teor. Veroyatnost. i Primenen 23 (1978), 358–375.Google Scholar
(1)Barndorff-Nielsen, O.On the limit behaviour of extreme order statistics. Ann. Math. Statist. 34 (1963), 992–1002.CrossRefGoogle Scholar
(2) De Haan, L.On regular variation and its application to the weak convergence of sample extremes (Mathematical Centre Tracts 32, Amsterdam, 1970).Google Scholar
(3)De Haan, L. and Hordijk, A.he rate of growth of sample maxima. Ann. Math. Statist. 43 (1972), 1185–1196.CrossRefGoogle Scholar
(4)Eicker, F.A loglog-law for double sequences of random variables I. Z. Wahrscheinlich-keitstheorie verw. Gebiete 16 (1970), 107–133.CrossRefGoogle Scholar
(5)Gnedenko, B. V.Sur la distribution liinite du terme maximum d'une série aléatoire. Ann. of Math. 44 (1943), 423–453.CrossRefGoogle Scholar
(6)Hall, P.A note on a paper of Barnes and Tucker. J. London Math. Soc. 19 (1979), 170–174.CrossRefGoogle Scholar
(7)Kiefer, J.Iterated logarithm analogues for sample quantiles when pn ↓ 0. Proc. Sixth Berkeley Symp. 1 (1971), 227–244.Google Scholar
(8)Mejzler, D. G.On a theorem of B. V. Gnedenko. Trudov Inst. Mat. Akad. Nauk Ukrain. R.S.R. 12 (1949), 31–35 (in Russian).Google Scholar
(9)Pickands, J.Sample sequences of maxima. Ann. Math. Statist. 38 (1969), 1570–1574.Google Scholar
(10)Resnick, S. I. and Tomkins, R. J.Almost sure stability of maxima. J. Appl. Probability 10 (1973), 387–401.CrossRefGoogle Scholar
(11)Smirnov, N. V.Limit distributions for the terms of a variational series. Amer. Math. Soc. Transl. ser. 1, no. 67 (1952), 82–143.Google Scholar
(12)Smirnov, N. V.Some remarks on limit laws for order statistics. Theor. Probability Appl. 12 (1967), 337–339.Google Scholar
(13)Wichura, M. J.On the functional form of the law of the iterated logarithm for the partial maxima of independent identically distributed random variables. Ann. Probability 2 (1974), 202–230.CrossRefGoogle Scholar