A Strong Invariance Theorem of the Tail Empirical Copula Processes

We study the behavior of bivariate empirical copula process ${\mathbb{G}}_n(\cdot ,\cdot )$ on pavements $[0,k_n\mathrm{/}n{]}^2$ 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 $k_n\mathrm{/}n\searrow \gamma$ as $n\to \mathrm{\infty },$ where $0\le \gamma \le 1.$