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A law of the iterated logarithm for sums of extreme values from a distribution with a regularly varying upper tail. (English) Zbl 0646.60034

Let \(X_ 1\), \(X_ 2\),... be independent observations from a distribution with a regularly varying upper tail with index a greater than 2. For each \(n\geq 1\), let \(X_{1,n}\leq...\leq X_{n,n}\) denote the order statistics based on \(X_ 1\),..., \(X_ n\). Choose any sequence of integers \((k_ n)_{n\geq 1}\) such that \(1\leq k_ n\leq n\), \(k_ n\to \infty\), and \(k_ n/n\to 0.\)
It has been recently shown by S. Csörgö and D. M. Mason [Ann. Probab. 14, 974-983 (1986; Zbl 0593.60034)] that the sum of the extreme values \(X_{n,n}+...+X_{n-k_ n,n}\), when properly centered and normalized, converges in distribution to a standard normal random variable. In this paper, we completely characterize such sequences \((k_ n)_{n\geq 1}\) for which the corresponding law of the iterated logarithm holds.

MSC:

60F15 Strong limit theorems
62G30 Order statistics; empirical distribution functions

Citations:

Zbl 0593.60034
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