Multiple Extremal Integrals

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.

💡 Research Summary

The paper introduces a new class of stochastic objects called multiple extremal integrals, which extend the well‑known single extremal integral used extensively in extreme‑value theory. The construction starts from an independently scattered α‑Fréchet random sup measure (M_\alpha) defined on a σ‑finite measure space ((E,\mathcal{E},\mu)). For a non‑negative measurable function (f) on (E), the single extremal integral (\int_E f(u)M_\alpha(du)) is known to have an α‑Fréchet distribution with scale parameter (\bigl(\int_E f(u)^\alpha\mu(du)\bigr)^{1/\alpha}).

To generalize to higher dimensions, the authors consider the product space (E^k) and remove the diagonal set ({u_i=u_j}) (the “off‑diagonal” region). On this region they define a product‑type random sup measure (M^{(k)}_\alpha) by setting, for pairwise disjoint measurable sets (A_1,\dots,A_k), \