Bahadur--Kiefer Representations for Time Dependent Quantile Processes

TL;DR

This paper develops strong approximation results for time-dependent empirical processes based on fractional Brownian motions, using Bahadur--Kiefer representations, leading to laws of the iterated logarithm for quantile processes.

Contribution

It introduces time-dependent Bahadur--Kiefer representations for empirical processes driven by fractional Brownian motions, providing new strong approximation and limit theorems.

Findings

Established strong approximations to the empirical process by Gaussian processes.

Derived functional laws of the iterated logarithm for quantile processes.

Provided new insights into the asymptotic behavior of time-dependent empirical processes.

Abstract

We define a time dependent empirical process based on $n$ independent fractional Brownian motions and describe strong approximations to it by Gaussian processes. They lead to strong approximations and functional laws of the iterated logarithm for the quantile or inverse of this empirical process. They are obtained via time dependent Bahadur--Kiefer representations.