A Vector Space Approach to Heavy Tailed Analysis

Abstract

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 $b(s)=s^\alpha L(s)$, 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 $\alpha > 1$. 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 $\alpha = 2$, 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.