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Journal of the London Mathematical Society 1978 s2-18(1):181-191; doi:10.1112/jlms/s2-18.1.181
© 1978 by London Mathematical Society
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© Oxford University Press

On the Extreme Terms of a Sample From the Domain of Attraction of a Stable Law

Peter Hall

Department of Statistics, S.G.S., The Australian National University P.O. Box 4, Canberra, A.C.T. 2600, Australia

Received 23 August 1977. Received 6 February 1978.

Let X1, X2, ..., Xn be independent random variables having a common distribution in the domain of attraction of a stable law with exponent {alpha} < 2. Let {Xn1, Xn2, ..., Xnn} denote the sample {X1, X2, ..., Xn} arranged in order of decreasing magnitude:


Formula

It is known that the variables Formula and Formula, when suitably normalised, have a joint limiting distribution ((k)T, T(k)) as n -> {infty}. We find the characteristic function of T(k) and show that for suitable constants ck, (k)T–ck has a limiting stable distribution as k->{infty}. We derive a rate of convergence.


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