An Offspring of Multivariate Extreme-Value Theory: The Max-Characteristic Function

TL;DR

This paper introduces max-characteristic functions (max-CFs) as a new tool in multivariate extreme-value theory, characterizing distributions of nonnegative random vectors and relating to Wasserstein convergence.

Contribution

It defines max-CFs, proves their convergence properties, shows the space is not closed, and provides an inversion formula, advancing theoretical understanding in multivariate extremes.

Findings

Max-CFs characterize distributions of nonnegative vectors.

Pointwise convergence of max-CFs is equivalent to Wasserstein convergence.

The space of max-CFs is not closed under pointwise limits.

Abstract

This paper introduces max-characteristic functions (max-CFs), which are an offspring of multivariate extreme-value theory. A max-CF characterizes the distribution of a random vector in R^d , whose components are nonnegative and have finite expectation. Pointwise convergence of max-CFs is shown to be equivalent with convergence with respect to the Wasserstein metric. The space of max-CFs is not closed in the sense of pointwise convergence. An inversion formula for max-CFs is established.