## 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

Zbl 0593.60034
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