On the distributions of \(L_ p\) norms of weighted uniform empirical and quantile processes. (English) Zbl 0646.62015

Let \(U_{1,n}\leq U_{2,n}\leq...\leq U_{n,n}\) be the order statistics based on a random sample \(U_ 1,U_ 2,...,U_ n\) from a uniform (0,1) distribution. For \(0\leq s\leq 1\), define the uniform quantile process \(u_ n(s)\) as \(u_ n(0)=0\) and \(u_ n(s)=n^{1/2}(s-U_{k,n})\) if \((k-1)/n<s\leq k/n\), \(k=1,2,...,n\); also, for \(0\leq s\leq 1\), let \(e_ n(s)=n^{1/2}(E_ n(s)-s)\), where \(E_ n(.)\) is the empirical distribution function based on \(U_ 1,U_ 2,...,U_ n\). Asymptotic distributions of functionals of the form \[ \int_{A_ n}| u_ n(s)| \quad p/q(s)ds\quad and\quad \int_{A_ n}| e_ n(s)| \quad p/q(s)ds \] have been studied for various choices of intervals of integration \(A_ n\), where \(1\leq p<\infty\) and q(.) is a weight function assumed to be positive on (0,\()\) and regularly varying at zero, for the purposes of all the theorems except the first one which applies even when more generally q(s) is such that \(\int^{1/2}_{0}s^{p/2}/q(s)ds<\infty\).
Reviewer: S.K.Basu


62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
60F25 \(L^p\)-limit theorems
62G30 Order statistics; empirical distribution functions
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