Strong limit theorems for oscillation moduli of the uniform empirical process

Abstract

Let \(U_n (t) = n^{\tfrac{1}{2}} (\Gamma _n (t) - t), 0 \leqq t \leqq 1\), denote the uniform empirical process based on the first n of a sequence ξ 1, ξ 2, ... of iid uniform (0,1) random variables where \(\Gamma _n (t) = n^{ - 1} \sum\limits_{i = 1}^n {1_{[0,t]} } (\xi _i )\) is the empirical distribution function. The oscillation modulus of U n is defined by $$\omega _n (a) = sup \{ |U_n (t + h) - U_n (t)|: 0 \leqq t \leqq 1 - h, h \leqq a\}$$ , and the Lipschitz-1/2 modulus of U n is defined by $$\tilde \omega _n (a) = sup \{ |U_n (t + h) - U_n (t)|/h^{\tfrac{1}{2}} : 0 \leqq t \leqq 1 - h, a \leqq h \leqq 1\}$$