Extremal stochastic integrals: a parallel between max-stable processes and α-stable processes

Abstract

We construct extremal stochastic integrals R e E f ðuÞM ðduÞ of a deterministic function f ðuÞ ! 0 with respect to a random ÀFré chet ( > 0) sup-measure. The measure M is sup-additive rather than additive and is defined over a general measure space ðE; E; "Þ, where " is a deterministic control measure. The extremal integral is constructed in a way similar to the usual Àstable integral, but with the maxima replacing the operation of summation. It is well-defined for arbitrary f ðuÞ ! 0; R E f ðuÞ "ðduÞ < 1, and the metric & ð f ; gÞ : ¼ R E j f ðuÞ À gðuÞ j"ðduÞ metrizes the convergence in probability of the resulting integrals.This approach complements the well-known de Haan's spectral representation of max-stable processes with ÀFré chet marginals. De Haan's representation can be viewed as the max-stable analog of the LePage series representation of Àstable processes, whereas the extremal integrals correspond to the usual Àstable stochastic integrals. We prove that essentially any strictly Àstable process belongs to the domain of max-stable attraction of an ÀFré chet, max-stable process. Moreover, we express the corresponding ÀFré chet processes in terms of extremal stochastic integrals, involving the kernel function of the Àstable process. The close correspondence between the max-stable and Àstable frameworks yields new examples of max-stable processes with non-trivial dependence structures.

On the structure and representations of max-stable processes

“…[6] and [34]). In the sequel, we equip the space L α + (S, µ) with the metric ρ µ,α and often write f L α + (S,µ) for ( S f α dµ) 1/α .…”

“…It turns out that a stochastic process {X t } t∈T with α-Fréchet marginals is max-stable if and only if all positive max-linear combinations: max 1≤j ≤n a j X t j ≡ 1≤j ≤n a j X t j for all a j > 0, t j ∈ T , 1 ≤ j ≤ n, (2.1) are α-Fréchet random variables (see, e.g. [7] and [34]). This feature resembles the definition of Gaussian or, more generally, symmetric α-stable (sum-stable) processes, where all finitedimensional linear combinations are univariate Gaussian or symmetric α-stable, respectively (see, e.g.…”

“…Here, we adopt the slightly more general, but essentially equivalent, approach of representing max-stable processes through extremal integrals with respect to random supmeasures (see [34]). We do so in order to emphasize the analogies with the well-developed theory of sum-stable processes (see, e.g.…”

“…+ (S, µ). For more details, see [34]. Now, for any collection of deterministic functions {f t } t∈T ⊂ L α + (S, µ), we can construct the stochastic process In view of the max-linearity of the extremal integrals and (2.1), the resulting process X = {X t } t∈T is α-Fréchet.…”

Norm, Point, and Distance Estimation Over Multiple Signals Using Max-Stable Distributions

“…Due to the non-linearity of the max-stable sketches, this metric, rather than the norm f − g α , is more natural in our setting [22]. Suppose for example that we have indicator signals, i.e.…”