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Multiple Extremal Integrals

Shuyang Bai, Jiemiao Chen

TL;DR

This work develops a comprehensive theory of multiple extremal integrals $I_k^e(f)$ driven by independently scattered $\alpha$-Fréchet random sup measures. It introduces product random sup measures $M_\alpha^{(k)}$ on off-diagonal spaces, provides a LePage-type representation, and derives integrability, convergence, tail behavior, and independence results, including a product formula and max-infinitely divisible structure. The authors also extend extreme-value models to a multi-regenerative setting, unveiling phase transitions in extremal clustering and macroscopic limits governed by the interaction of renewal sets. Collectively, the results advance multi-argument extreme value theory, offering a rigorous framework for analyzing joint extremes, dependence, and clustering in high-order extremal integrals and related stationary models.

Abstract

We introduce the notion of multiple extremal integrals as an extension of single extremal integrals, which have played important roles in extreme value theory. The multiple extremal integrals are formulated in terms of a product-form random sup measure derived from the $α$-Fréchet random sup measure. We establish a LePage-type representation similar to that used for multiple sum-stable integrals, which have been extensively studied in the literature. This approach allows us to investigate the integrability, tail behavior, and independence properties of multiple extremal integrals. Additionally, we discuss an extension of a recently proposed stationary model that exhibits an unusual extremal clustering phenomenon, now constructed using multiple extremal integrals.

Multiple Extremal Integrals

TL;DR

This work develops a comprehensive theory of multiple extremal integrals driven by independently scattered -Fréchet random sup measures. It introduces product random sup measures on off-diagonal spaces, provides a LePage-type representation, and derives integrability, convergence, tail behavior, and independence results, including a product formula and max-infinitely divisible structure. The authors also extend extreme-value models to a multi-regenerative setting, unveiling phase transitions in extremal clustering and macroscopic limits governed by the interaction of renewal sets. Collectively, the results advance multi-argument extreme value theory, offering a rigorous framework for analyzing joint extremes, dependence, and clustering in high-order extremal integrals and related stationary models.

Abstract

We introduce the notion of multiple extremal integrals as an extension of single extremal integrals, which have played important roles in extreme value theory. The multiple extremal integrals are formulated in terms of a product-form random sup measure derived from the -Fréchet random sup measure. We establish a LePage-type representation similar to that used for multiple sum-stable integrals, which have been extensively studied in the literature. This approach allows us to investigate the integrability, tail behavior, and independence properties of multiple extremal integrals. Additionally, we discuss an extension of a recently proposed stationary model that exhibits an unusual extremal clustering phenomenon, now constructed using multiple extremal integrals.

Paper Structure

This paper contains 27 sections, 46 theorems, 274 equations.

Table of Contents

  1. Introduction
  2. Multiple extremal integrals: constructions and basic properties
  3. alpha-Fréchet random sup measures
  4. Product random sup measures
  5. Multiple extremal integrals
  6. Alternative construction via Poisson point process
  7. Integrability
  8. Sufficient conditions
  9. Necessary condition
  10. A discussion on the moment-type conditions
  11. Convergence of multiple extremal integrals
  12. Tail behavior
  13. Independence between multiple extremal integrals
  14. Product formula
  15. Tail of the product and pairwise independence of extremal integrals
  16. ...and 12 more sections

Key Result

Theorem 2.5

Suppose $\mu$ is a finite measure. For any $A \in \mathcal{E}^{(k)}$, $k\in \mathbb{Z}_+$, there exists a sequence $(A_n)_{n \in \mathbb{Z}_+}$, with $A_n \in \mathcal{F}_k, n\in \mathbb{Z}_+$, such that $\mu^k(A_n \Delta A) \rightarrow 0$ as $n\rightarrow\infty$. In addition, the $L^\gamma$ limit o Further, for any $\gamma\in (0,\alpha)$ and $r>\alpha$, there is a constant $c>0$ depending on $(r,

Theorems & Definitions (106)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Proposition 2.8
  • ...and 96 more