Besov-Orlicz moduli of Brownian motion and polygonal partial sum processes

TL;DR

This paper characterizes the exact Besov-Orlicz regularity of Brownian motion and polygonal partial sum processes, extending regularity results and strengthening invariance principles in Banach spaces.

Contribution

It introduces a new chaining bound for the Orlicz modulus, extends Donsker's invariance principle, and applies these results to nonparametric statistical testing.

Findings

Exact limit of the sub-Gaussian Orlicz modulus for Brownian motion in Banach spaces.

A new chaining bound for the Orlicz modulus of stochastic processes.

Strengthened invariance principle for polygonal partial sum processes.

Abstract

The sample paths of Brownian motion are known to admit the exact Besov-type smoothness exponent 1/2 when measured in the sub-Gaussian Orlicz norm. We extend these regularity results by deriving the exact limit of the sub-Gaussian Orlicz modulus for Brownian motion in Banach spaces, and we provide a rate of convergence towards this limiting value. The central technique is a new chaining bound for the Orlicz modulus of a stochastic process. The latter also applies to polyogonal partial sum processes of functional random variables and allows us to strengthen Donsker's invariance principle to all function spaces on the Besov-Orlicz scale up to the exact modulus with exponent 1/2. For the critical case, we establish the thresholded weak convergence of the Besov-Orlicz seminorm of the partial sum process. The analytical results find application in a nonparametric statistical testing problem, where Besov-Orlicz statistics are shown to detect a broader range of alternatives compared to H\"olderian multiscale statistics.