On bivariate lower semilinear copulas and the star product

We revisit the family $\mathcal{C}^{LSL}$ of all bivariate lower semilinear (LSL) copulas first introduced by Durante et al. in 2008 and, using the characterization of LSL copulas in terms of diagonals with specific properties, derive several novel and partially unexpected results. In particular we prove that the star product (also known as Markov product) $S_{\delta_1}*S_{\delta_2}$ of two LSL copulas $S_{\delta_1},S_{\delta_2}$ is again a LSL copula, i.e., that the family $\mathcal{C}^{LSL}$ is closed with respect to the star product. Moreover, we show that translating the star product to the class of corresponding diagonals $\mathcal{D}^{LSL}$ allows to determine the limit of the sequence $S_\delta, S_\delta*S_\delta, S_\delta*S_\delta*S_\delta,\ldots$ for every diagonal $\delta \in \mathcal{D}^{LSL}$. In fact, for every LSL copula $S_\delta$ the sequence $(S_\delta^{*n})_{n \in \mathbb{N}}$ converges to some LSL copula $S_{\overline{\delta}}$, the limit $S_{\overline{\delta}}$ is idempotent, and the class of all idempotent LSL copulas allows for a simple characterization. Complementing these results we then focus on concordance of LSL copulas. After deriving simple formulas for Kendall's $\tau$ and Spearman's $\rho$ we study the exact region $\Omega^{LSL}$ determined by these two concordance measures of all elements in $\mathcal{C}^{LSL}$, derive a sharp lower bound and finally show that $\Omega^{LSL}$ is convex and compact.