最大稳定分布的凸几何
Convex geometry of max-stable distributions
摘要：
证明了$[0,\infty)^d$中具有单位Fréchet边缘的最大稳定随机向量与$[0,\infty)^d$中的凸集$K$一一对应，称之为最大稳定集.的最大-zeros可以被描述为集作为交叉多面体的闵可夫斯基和的限制，或者，作为随机交叉多面体的选择期望，其分布由最大稳定随机向量的谱测度控制。此外，具有单位Fréchet边缘的最大稳定随机向量$ξ$的累积分布函数$\Prob{ξ\leq x}$由$x$的逆的范数确定，其中所有可能的范数由max-zerk的支撑函数给出。作为一个应用，几何解释的一些著名的概念，从多元极值理论和Copula提供。凸几何方法使得引入具有最大稳定随机向量的新运算成为可能。
英文摘要：
It is shown that max-stable random vectors in $[0,\infty)^d$ with unit Fréchet marginals are in one to one correspondence with convex sets $K$ in $[0,\infty)^d$ called max-zonoids. The max-zonoids can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or, alternatively, as the selection expectation of a random cross-polytope whose distribution is controlled by the spectral measure of the max-stable random vector. Furthermore, the cumulative distribution function $\Prob{ξ\leq x}$ of a max-stable random vector $ξ$ with unit Fréchet marginals is determined by the norm of the inverse to $x$, where all possible norms are given by the support functions of max-zonoids. As an application, geometrical interpretations of a number of well-known concepts from the theory of multivariate extreme values and copulas are provided. The convex geometry approach makes it possible to introduce new operations with max-stable random vectors.
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