A Strong Invariance Theorem of the Tail Empirical Copula Processes

TL;DR

This paper establishes a strong invariance theorem for the tail empirical copula process, providing bounds for its approximation by Gaussian processes in the context of bivariate distributions.

Contribution

It introduces a novel invariance result for tail empirical copula processes, extending understanding of their asymptotic behavior under specific conditions.

Findings

Provides an upper bound for the strong approximation by Gaussian processes.

Extends tail empirical copula process theory to new asymptotic regimes.

Offers insights into the behavior of copula processes on tail regions.

Abstract

We study the behavior of bivariate empirical copula process $\mathbb{G}_n(\cdot,\cdot)$ on pavements $[0,k_n/n]^2$ of $[0,1]^2,$ where $k_n$ is a sequence of positive constants fulfilling some conditions. We provide a upper bound for the strong approximation of $\mathbb{G}_n(\cdot,\cdot)$ by a Gaussian process when $k_n/n \searrow \gamma$ as $n\rightarrow \infty,$ where $0 \leq \gamma \leq 1.$