A necessary and sufficient condition for convergence in distribution of the quantile process in $L^1(0,1)$

TL;DR

This paper provides a precise criterion for the convergence in distribution of the quantile process in L^1(0,1), linking it to properties of the quantile function and bootstrap approximation.

Contribution

It establishes a necessary and sufficient condition for convergence of the quantile process in L^1(0,1), connecting function regularity with probabilistic convergence.

Findings

Convergence occurs if the quantile function is locally absolutely continuous.

A slight strengthening of square integrability is required for convergence.

Bootstrap methods can approximate the quantile process if convergence occurs.

Abstract

We establish a necessary and sufficient condition for the quantile process based on iid sampling to converge in distribution in $L^1(0,1)$. The condition is that the quantile function is locally absolutely continuous and satisfies a slight strengthening of square integrability. If the quantile process converges in distribution then it may be approximated using the bootstrap.