Journal of Applied Mathematics and Stochastic Analysis
Volume 7 (1994), Issue 2, Pages 125-143
doi:10.1155/S1048953394000146
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.