Besov regularity of the uniform empirical process

TL;DR

This paper investigates the Besov regularity of the continuous uniform empirical process paths within specific sequence spaces, extending known results from Brownian motion and bridge to empirical processes.

Contribution

It establishes the regularity properties of the uniform empirical process in Besov and sequence spaces, which was previously studied mainly for Brownian motion.

Findings

Regularity properties are established for the empirical process in Besov spaces.

Results extend known regularity results from Brownian motion to empirical processes.

Analysis includes specific subspaces for the empirical process paths.

Abstract

The paths of Brownian motion have been widely studied in the recent years relatively in Besov spaces $B_{p, \infty}^\a$. The results are the same as to the Brownian bridge. In fact these regularities properties are established in some sequence spaces $S_{p, \infty}^\a$ using an isomorphisim between them and $B_{p, \infty}^\a$. In this note, we are concerned with the regularity of the paths of the continuous version of the uniform empirical process in the space $S_{p, \infty}^\a$ and in one of his separable sub space $S_{p, \infty}^{\a, 0}$ for a suitable choice of $\a$ and $p$.