一种重尾分析的向量空间方法
A Vector Space Approach to Heavy Tailed Analysis
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中文总结 AI 辅助
该研究构建基于随机变量单变量正则变化的向量空间，定义相关子空间与等价类得向量空间$\mathbb{W}_b$，证其性质，还用于极值建模，如在$\alpha = 2$时相关度量可作内积并利用投影定理线性预测。
中文摘要 AI 辅助
我们构建了一个向量空间，其定义特征源于随机变量的单变量正则变化。具体而言，基础向量空间$\mathbb{V}_b$由这样的随机变量组成，其极限尾概率在通过形如$b(s)=s^\alpha L(s)$的正则变化函数缩放后是有限的。定义一个子空间${\cal N}_b$，对应于$\mathbb{V}_b$中经$b(s)$归一化后极限尾概率为零的随机变量，这使得基础空间$\mathbb{V}_b$能被划分为等价类。我们定义了一个由这些等价类组成的向量空间$\mathbb{W}_b$，并表明其非零元素是正则变化随机变量的等价类。我们证明了若$\alpha > 1$，$\mathbb{W}_b$存在自然范数。还表明等价类和范数收敛与更常见的随机变量向量空间不同。在极值建模方面，考虑$\mathbb{W}_b$的有限维子空间，其基向量联合正则变化。证明在$\alpha = 2$的情况下，先前定义的尾成对依赖度量可作为内积。由于任何有限维空间是完备的，可利用投影定理进行线性预测。
英文摘要
We construct a vector space whose defining characteristics are rooted in univariate regular variation of random variables. Specifically, the base vector space $\mathbb{V}_b$ consists of random variables whose limiting tail probabilities, when scaled by regularly varying functions of the form $b(s)=s^αL(s)$, are finite. Defining a subspace ${\cal N}_b$ corresponding to random variables in $\mathbb{V}_b$ whose limiting tail probabilities are zero when normalized by $b(s)$ allows the base space $\mathbb{V}_b$ to be partitioned into equivalence classes. We define a vector space $\mathbb{W}_b$ consisting of these equivalence classes, and show its nonzero elements are equivalence classes of regularly varying random variables. We show that a natural norm exists for $\mathbb{W}_b$ if $α> 1$. We show that the equivalence classes and convergence in norm are different than more familiar vector spaces of random variables. Turning our attention to extreme value modeling, we consider finite-dimensional subspaces of $\mathbb{W}_b$ whose basis vectors are jointly regularly varying. We show that in the case $α= 2$, the previously defined tail pairwise dependence measure serves as an inner product. As any finite-dimensional space is complete, we can use the projection theorem to perform linear prediction.
