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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 31, Fasc. 2,
pages 285 - 299
 

THE ASYMPTOTICS OF L-STATISTICS FOR NON I.I.D. VARIABLES WITH HEAVY TAILS

Adam Barczyk
Arnold Janssen
Markus Pauly

Abstract: The purpose of this paper is to study the asymptotic behaviour of linear combinations of order statistics (L -statistics)

     ∑kn
Ln :=    ci,nXi:kn
     i=1
with real scores ci,n  for variables with heavy tails. The order statistics Xi:k
   n  correspond to a non i.i.d. triangular array (Xi,n)1≤i≤k
         n  of infinitesimal and rowwise independent random variables. We give sufficient conditions for the convergence of L-statistics to non-normal limit laws and it is shown that only the extremes contribute to the limit distribution, whereas the middle parts vanish. As an example we consider the case, where the extremal partial sums belong to the domain of attraction of a stable law. We also study L-statistics with scores defined by ci,n := J(i∕(n + 1)) with a regularly varying function J , a case which has often been treated in the literature.

2000 AMS Mathematics Subject Classification: Primary: 62G32; Secondary: 62G30, 60F05.

Keywords and phrases: L-statistics, shift-compactness, order statistics, heavy tails, stable law, non i.i.d. array.

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