## 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

### 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
Full Text: