2026/07/20 by Kenneth Broadhead, Daniel Cooley
#math.PR #math.ST #stat.TH
We construct a vector space whose defining characteristics are rooted in univariate regular variation of random variables. Specifically, the base vector space \mathbbVb consists of random variables whose limiting tail probabilities, when scaled by regularly varying functions of the form b(s)=sαL(s), are finite. Defining a subspace \cal Nb corresponding to random variables in \mathbbVb whose limiting tail probabilities are zero when normalized by b(s) allows the base space \mathbbVb to be partitioned into equivalence classes. We define a vector space \mathbbWb 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 \mathbbWb 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 \mathbbWb 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.