# Weak convergence of quantile and expectile processes under general assumptions

## TL;DR

This paper establishes weak convergence of quantile and expectile processes to Gaussian limits under broad, less restrictive conditions, and confirms bootstrap consistency for these processes.

## Contribution

It introduces new weak convergence results for quantile and expectile processes under general assumptions, expanding applicability beyond traditional norms.

## Key findings

- Weak convergence to Gaussian processes in a specialized function space
- Expectiles require only finite second moments, no smoothness assumptions
- Bootstrap consistency for the convergence mode

## Abstract

We show weak convergence of quantile and expectile processes to Gaussian limit processes in the space of bounded functions endowed with an appropriate semimetric which is based on the concepts of epi- and hypo convergence as introduced in \citet{buecher2014}. We impose assumptions for which it is known that weak convergence with respect to the supremum norm or the Skorodhod metric generally fails to hold. For expectiles, we only require a distribution with finite second moment but no further smoothness properties of distribution function, for quantiles, the distribution is assumed to be absolutely continuous with a version of its Lebesgue density which is strictly positive and has left- and right-sided limits. We also show consistency of the bootstrap for this mode of convergence.