标题： 极端依赖概念 显示英文标题

标题： Extremal Dependence Concepts

摘要： 随机变量之间关系的概率表征需要依赖性的概念。 依赖性建模带来了数学和统计上的挑战，极端依赖概念的最新发展在概率及其在多个学科中的应用中引起了广泛关注。 本文的目的是回顾各种极端正相关和负相关的概念，包括一些最近建立的结果，重建它们的历史，将其与概率优化问题联系起来，并提供该领域的一系列开放问题。 虽然对于任意维度的随机向量来说，极端正相关的概念是达成一致的，但当涉及多于两个随机变量时，会出现多种极端负相关的概念。 我们回顾了文献中现有的流行的极端负相关概念，并引入了一个新的概念，该概念在一般意义上包含了现有的概念作为特例。 尽管许多关于依赖性的文献集中在正相关上，但我们表明负相关在许多优化问题的求解中同样重要。 虽然目前用来建模依赖性的最常用工具是Copula函数，但在本文中我们使用了等价的概念，即重排集合。 这不仅出于历史原因。 重排函数以完全确定的方式描述了随机变量之间的关系，有助于更深入地理解依赖性本身，并在广泛的一类优化问题的解的近似中具有诸多优势。 显示英文摘要

摘要： The probabilistic characterization of the relationship between two or more random variables calls for a notion of dependence. Dependence modeling leads to mathematical and statistical challenges, and recent developments in extremal dependence concepts have drawn a lot of attention to probability and its applications in several disciplines. The aim of this paper is to review various concepts of extremal positive and negative dependence, including several recently established results, reconstruct their history, link them to probabilistic optimization problems, and provide a list of open questions in this area. While the concept of extremal positive dependence is agreed upon for random vectors of arbitrary dimensions, various notions of extremal negative dependence arise when more than two random variables are involved. We review existing popular concepts of extremal negative dependence given in literature and introduce a novel notion, which in a general sense includes the existing ones as particular cases. Even if much of the literature on dependence is focused on positive dependence, we show that negative dependence plays an equally important role in the solution of many optimization problems. While the most popular tool used nowadays to model dependence is that of a copula function, in this paper we use the equivalent concept of a set of rearrangements. This is not only for historical reasons. Rearrangement functions describe the relationship between random variables in a completely deterministic way, allow a deeper understanding of dependence itself, and have several advantages on the approximation of solutions in a broad class of optimization problems.