Precise Asymptotics for the Uniform Empirical Process and the Uniform Sample Quantile Process

Abstract

One of the sources of “invariance principle” is that the limit properties of the uniform empirical process coincide with that of a Brownian bridge. The deep discussion of limit theorem of the uniform empirical process gathered wild interest of the researchers. In this paper, the precise convergence rate of the uniform empirical process is considered. As is well-known, when $\epsilon$ tends to 0, the precise asymptotic theorems can be demonstrated by referring to the classical method of Gut and Spǎtaru, by using some nice probability inequalities and so on. However, if $\epsilon$ tends to a positive constant, other powerful methods and tools are needed. The method of strong approximation is used in this paper. The main theorems are proved by using the Brownian bridge $B\left(t\right)$ to approximate the uniform empirical process ${\alpha }_n\left(t\right)$ . The relevant results for the uniform sample quantile process are also presented.