Symmetry of functions and exchangeability of random variables

Abstract

We present a new approach for measuring the degree of exchangeability of two continuous, identically distributed random variables or, equivalently, the degree of symmetry of their corresponding copula. While the opposite of exchangeability does not exist in probability theory, the contrary of symmetry is quite obvious from an analytical point of view. Therefore, leaving the framework of probability theory, we introduce a natural measure of symmetry for bivariate functions in an arbitrary normed function space. Restricted to the set of copulas this yields a general concept for measures of (non-)exchangeability of random variables. The fact that copulas are never antisymmetric leads to the notion of maximal degree of antisymmetry of copulas. We illustrate our approach by various norms on function spaces, most notably the Sobolev norm for copulas.