A Strong Invariance Theorem of the Tail Empirical Copula Processes

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 $n\to \infty ,$ where $0\le \gamma \le 1.$