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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 10, Fasc. 2,
pages 205 - 222
 

THE DOMAIN OF ATTRACTION OF STABLE LAWS AND EXTREME ORDER STATISTICS

Arnold Janssen

Abstract: The purpose of this paper is to study the asymptotic behaviour of trimmed sums of order statistics

(k sum (n) n- sum r ) Yn = a-n 1 Xi:n + Xi:n - bn, i=k i=n+1-r(n)
where the order statistic Xi:n arises from an i.i.d. sequence belonging to the domain of attraction of a stable law with index 0 < a < 2. If we use a special representation for Xi:n related to - 1 F (Ui:n) coming from uniformly distributed random variables U1,...,Un, then we can prove the convergence in probability or even 1 L -convergence for Yn in various cases. Special attention is devoted to the convergence of Yn to one-sided stable laws showing that we may choose min(k(n),r(n)) = 0. As an example we obtain the limiting distribution of student’s t type statistics.

2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;

Key words and phrases: -

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