Concordance, symmetrization and non-exchangeability for bivariate copulas

We study the relationship between measures of non-exchangeability $\mu_p$ ($p\in[1,+\infty]$), in the sense of Durante et al.

(2010), and classical dependence functionals for bivariate copulas.

We show that the symmetrization $C\mapsto(C+C^t)/2$ preserves Spearman's $\rho$ while annihilating $\mu_p$, and that Blomqvist's $\beta$ carries no information about the degree of non-exchangeability.

We also establish the sharp lower bound $\sigma(C)\ge 6\,\mu_1(C)$, where $\sigma$ is the Schweizer-Wolff dependence measure, showing that asymmetry implies dependence.

Closed-form expressions for $\tau$, $\rho$, and the tail-dependence coefficients of the maximally non-exchangeable family $\{M_\theta\}$ are derived as illustrations.