On joint sum/max stability and sum/max domains of attraction

TL;DR

This paper develops a new theoretical framework using harmonic analysis on semigroups to characterize the limit distributions of joint sum and maximum of dependent i.i.d. random vectors, extending classical results to dependent cases.

Contribution

It introduces a novel approach employing harmonic analysis on semigroups to analyze joint sum/max stability and domains of attraction with dependence.

Findings

Characterizes limit distributions for dependent sum and maximum.

Extends classical sum/max stability theory to dependent variables.

Provides a mathematical framework for describing domains of attraction.

Abstract

Let (W_i, J_i) be a sequence of i.i.d. R_+ x R-valued random vectors. Considering the partial sum of the first component and the corresponding maximum of the second component, we are interested in the limit distributions that can be obtained under an appropriate scaling. In the case that W_i and J_i are independent, the joint distribution of the sum and the maximum is the product measure of the limit distributions of the two components. But if we allow dependence between the two components, this dependence can still appear in the limit, and we need a new theory to describe the possible limit distributions. This is achieved via harmonic analysis on semigroups, which can be utilized to characterize the scaling limit distributions and describe their domains of attraction.