标题： 关于双变量下半线性copula和星积 显示英文标题

标题： On bivariate lower semilinear copulas and the star product

摘要： 我们重新研究了由Durante等人于2008年首次引入的所有二元下半线性（LSL）copulas的族$\mathcal{C}^{LSL}$，并利用LSL copulas在对角线上的特性表征，推导出若干新颖且部分意外的结果。特别是我们证明了两个LSL copulas$S_{\delta_1},S_{\delta_2}$的星积（也称为马尔可夫积）$S_{\delta_1}*S_{\delta_2}$仍然是一个LSL copula，即该族$\mathcal{C}^{LSL}$对星积是封闭的。此外，我们表明将星积转换到相应的对角线类$\mathcal{D}^{LSL}$允许确定序列$S_\delta, S_\delta*S_\delta, S_\delta*S_\delta*S_\delta,\ldots$对于每个对角线$\delta \in \mathcal{D}^{LSL}$的极限。 事实上，对于每个LSL陪集$S_\delta$，序列$(S_\delta^{*n})_{n \in \mathbb{N}}$收敛到某个LSL陪集$S_{\overline{\delta}}$，极限$S_{\overline{\delta}}$是幂等的，并且所有幂等的LSL陪集的类允许进行简单的表征。在补充这些结果之后，我们随后关注LSL陪集的一致性。 在推导出Kendall的$\tau$和Spearman的$\rho$的简单公式后，我们研究由所有元素在$\mathcal{C}^{LSL}$中的这两个同调度量确定的精确区域$\Omega^{LSL}$，推导出一个紧的下界，并最终证明$\Omega^{LSL}$是凸且紧的。 显示英文摘要

摘要： 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.