Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 2, Pages 125-143
doi:10.1155/S1048953394000146
Laws of large numbers for L-statistics
Institute of Mathematics and Informatics, Akademijos 4, Vilnius 2600, Lithuania
Received 1 February 1992; Revised 1 April 1994
Copyright © 1994 Rimas Norvaiša. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Consider Ln=n−1∑1≤i≤ncnig(Xn:i) for order statistics Xn:i
and let cni=n∫(i−1)/ni/nJdλ for some (Lebesgue) λ-summable over (0,1)
function J. Sufficient as well as necessary conditions for limnLn=∫01Jgdλ to hold almost surely and in probability are given. Superposition
(or Nemytskii) operators have been used to derive the laws of large
numbers for L-statistics from the laws of large numbers in quasi-Banach
function spaces for the empirical distribution functions based on
X1,…,Xn.