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

Csörgö, Miklós; Horváth, Lajos

Ann. Probab. 16, No. 1, 142-161 (1988).

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

MSC:

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

Keywords:

weak approximations; Ornstein-Uhlenbeck process; Brownian bridge; order statistics; uniform quantile process; empirical distribution; Asymptotic distributions of functionals; regularly varying at zero