On bivariate lower semilinear copulas and the star product

We revisit the family CLSL 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) Sδ₁*Sδ₂ of two LSL copulas Sδ₁,Sδ₂ is again a LSL copula, i.e., that the family CLSL is closed with respect to the star product. Moreover, we show that translating the star product to the class of corresponding diagonals DLSL allows to determine the limit of the sequence S_δ, S_δ*S_δ, S_δ*S_δ*S_δ,… for every diagonal δ ∈ DLSL. In fact, for every LSL copula S_δ the sequence (S_δ*n)n ∈ N converges to some LSL copula Sδ̄, the limit Sδ̄ 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 τ and Spearman's ρ we study the exact region ΩLSL determined by these two concordance measures of all elements in CLSL, derive a sharp lower bound and finally show that ΩLSL is convex and compact.