Couplings and Strong Approximations to Time Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions

TL;DR

This paper develops Gaussian couplings and strong approximations for a time-dependent empirical process based on i.i.d. fractional Brownian motions, leading to new laws of the iterated logarithm for such processes.

Contribution

It introduces a novel approach to approximate time-dependent empirical processes with Gaussian processes, extending classical results to fractional Brownian motions.

Findings

Established Gaussian couplings for the process

Derived strong approximation results

Proved functional laws of the iterated logarithm

Abstract

We define a time dependent empirical process based on $n$ i.i.d.~fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for this process.