The topological space of Schur-concave copulas is homeomorphic to the Hilbert cube

Abstract

Schur-concave copulas, a distinct subclass of copulas, have emerged as a crucial component in recent advancements of copula theory. In this paper, we employ techniques from infinite-dimensional topology to investigate the topological structure of the space of all Schur-concave copulas equipped with the uniform metric and its topological position in the (symmetric) copula space. Moreover, we show that there exists a homeomorphism from the copula space onto the Hilbert cube, which relates the symmetric copula space onto an end of the Hilbert cube and relates the Schur-concave copula space onto an end of the end of the Hilbert cube. As a consequence, the Schur-concave copula space has the fixed point property. Furthermore, we show that a subset of the family of all associative copulas is a Z-set in the Schur-concave copula space if and only if it is compact.