On Strong Approximation of the Multidimensional Empirical Process

Abstract

Let $\mathbf{X}_1, \mathbf{X}_2, \cdots$ be a sequence of i.i.d. rv's uniformly distributed over the unit square $I^2$. Further, let $F_n$ be the empirical distribution function based on the sample $\mathbf{X}_1, \mathbf{X}_2, \cdots, \mathbf{X}_n$. A sequence $\{B_n\}$ of Brownian bridges and a Kiefer process $K$ is constructed such that $$\sup_{A\in Q} |n^{\frac{1}{2}}(F_n(A) - \lambda(A)) - B_n(A)| = O(n^{-\frac{1}{19}}) \\ \sup_{A\in Q} |n(F_n(A) - \lambda(A)) - K(A; n)| = O(n\frac{1}{2}\frac{2}{5})$$ a.s. where $F_n(A), B_n(A), K(A; n)$ are the corresponding random measures of $A, \lambda$ is the Lebesgue measure and $Q$ is the set of Borel sets of $\mathrm{I}^2$ having twice differentiable boundaries.