Extreme-value copulas associated with the expected scaled maximum of independent random variables

TL;DR

This paper explores the connection between extreme-value copulas and the expected scaled maximum of independent random variables, introducing a new stochastic process representation and an efficient simulation algorithm for high-dimensional cases.

Contribution

It establishes a bijection between distribution functions and Lévy measures, linking well-known copulas like Gumbel and Galambos to a unified stochastic process framework.

Findings

Derived a bijection between distribution functions and Lévy measures.

Showed that certain copulas arise as margins of an exchangeable sequence.

Developed an efficient simulation algorithm for high-dimensional models.

Abstract

It is well-known that the expected scaled maximum of non-negative random variables with unit mean defines a stable tail dependence function associated with some extreme-value copula. In the special case when these random variables are independent and identically distributed, min-stable multivariate exponential random vectors with the associated survival extreme-value copulas are shown to arise as finite-dimensional margins of an infinite exchangeable sequence in the sense of De Finetti's Theorem. The associated latent factor is a stochastic process which is strongly infinitely divisible with respect to time, which induces a bijection from the set of distribution functions F of non-negative random variables with finite mean to the set of L\'evy measures on the positive half-axis. Since the Gumbel and the Galambos copula are the most popular examples of this construction, the investigation of this bijection contributes to a further understanding of their well-known analytical similarities. Furthermore, a simulation algorithm based on the latent factor representation is developed, if the support of F is bounded. Especially in large dimensions, this algorithm is efficient because it makes use of the De Finetti structure.