Improvement of two Hungarian bivariate theorems

TL;DR

This paper introduces a new technique to improve the error bounds in multivariate strong approximation theorems for empirical processes, achieving tighter bounds and identifying martingales in the error terms.

Contribution

It presents a novel method using martingale identification and exponential inequalities to refine error bounds in multivariate empirical process approximations.

Findings

Error bound for bivariate empirical process approximation is of order n^{-1/2}(log n)^{3/2}

Improved global error bounds for univariate and multivariate empirical processes

Proposed method likely optimal for d-variate case

Abstract

We introduce a new technique to establish Hungarian multivariate theorems. In this article we apply this technique to the strong approximation bivariate theorems of the uniform empirical process. It improves the Komlos, Major and Tusn\'ady (1975) result, as well as our own (1998). More precisely, we show that the error in the approximation of the uniform bivariate $n$-empirical process by a bivariate Brownian bridge is of order $n^{-1/2}(log (nab))^{3/2}$ on the rectangle $[0,a]x[0,b]$, $0 <a, b <1$, and that the error in the approximation of the uniform univariate $n$-empirical process by a Kiefer process is of order $n^{-1/2}(log (na))^{3/2}$ on the interval $[0,a]$, $0 < a < 1$. In both cases, the global error bound is therefore of order $n^{-1/2}(log (n))^{3/2}$. Previously, from the 1975 article of Komlos, Major and Tusn\'ady, the global error bound was of order $n^{-1/2}(log (n))^{2}$, and from our 1998 article, the local error bounds were of order $n^{-1/2}(log (nab))^{2}$ or $n^{-1/2}(log (na))^{2}$. We think that, in the d-variate case, the global error bound between the uniform d-variate $n$-empirical process and the associated Gaussian process is of order $n^{-1/2}(log (n))^{(d+1)/2}$, and that this result is optimal. The new feature of this article is to identify martingales in the error terms and to apply to them an exponential inequality. The idea is to bound of the compensator of the error term, instead of bounding of the error term itself.