When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

TL;DR

This paper introduces a new metric for analyzing the weak convergence of empirical processes in dependence functions and residuals, especially when traditional uniform convergence fails, and develops a related convergence theory.

Contribution

It proposes a novel locally bounded function metric and establishes weak convergence results for empirical processes where uniform convergence does not hold.

Findings

Weak convergence achieved under the new metric for complex dependence processes.

The new convergence implies local uniform convergence for continuous limits.

Applications include asymptotic analysis of resampling and goodness-of-fit tests.

Abstract

In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in case the underlying distributions lack a certain degree of smoothness. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epi- and hypo-convergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies $L^p$ convergence. For the examples mentioned above, weak convergence with respect to the new metric is established in situations where it does not occur with respect to the supremum distance. The results are applied to obtain asymptotic properties of resampling procedures and goodness-of-fit tests.