Gaussian Approximations and Related Questions for the Spacings process

TL;DR

This paper investigates the limitations of existing Gaussian approximation methods for the k-spacings process, providing new bounds, a Glivenko-Cantelli theorem, and extending results to the spacings process.

Contribution

It demonstrates that the traditional approach cannot surpass a specific rate and extends key empirical process results to the spacings process, offering new theoretical insights.

Findings

Established a lower bound on approximation rates.

Extended Glivenko-Cantelli theorem to spacings process.

Derived bounds for fixed and increasing k.

Abstract

All the available results on the approximation of the k-spacings process to Gaussian processes have only used one approach, that is the Shorack and Pyke's one. Here, it is shown that this approach cannot yield a rate better than $% \left( N/\log \log N\right) ^{-\frac{1}{4}}\left( \log N\right) ^{\frac{1}{2}% }$. Strong and weak bounds for that rate are specified both where k is fixed and where $k\rightarrow +\infty $. A Glivenko-Cantelli Theorem is given while Stute's result for the increments of the empirical process based on independent and indentically distributed random variables is extended to the spacings process. One of the Mason-Wellner-Shorack cases is also obtained.