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.
Copyright
Copyright © 1994 Hindawi Publishing Corporation. This is an open access article distributed under the
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