Reduction Principle for Functionals of Vector Random Fields

Abstract

We prove a version of the reduction principle for functionals of vector long-range dependent random fields. The components of the fields may have different long-range dependent behaviours. The results are illustrated by an application to the first Minkowski functional of the Fisher-Snedecor random fields. Simulation studies confirm the obtained theoretical results and suggest some new problems.Keywords excursion set · long-range dependence · first Minkowski functional · Fisher-Snedecor random fields · heavy-tailed · non-central limit theorems · random field · sojourn measure 1 IntroductionOver the last four decades, a great deal of effort has been devoted to studying the geometric characteristics of excursion sets of random fields. The obtained theoretical results have been utilised in a variety of applications, including in geoscience, astrophysics, medical imaging and other related fields (see Azaïs and Wschebor 2009). Among numerous stochastic models, Gaussian and related fields (such as χ 2 , F and t fields) are the most popular in studying excursion sets. The reason for this popularity is their simplicity and mathematical tractability.

Reduction principle for functionals of strong-weak dependent vector random fields

“…This approach can be extended to the case of random fields as Y (x) = η 1 (x)+ η2 (x), x ∈ R d , resulting in Y (x) with skewed marginal distributions. In the example below we use η2 (x) = η 2 2 (x) and show that contrary to the reduction principle for strongly dependent vector random fields in Olenko and Omari (2019)…”

“…The random variable K * r,l ≡ 0 if and only if l ∈ L + . Theorem 1 in Olenko and Omari (2019) gives a reduction principle for vector random fields with strongly dependent components. The following result complements it for the case of random fields with strongly and weakly dependent components.…”

“…The classical non-central limit theorems are based on another "proper" reduction principle, when the asymptotic behaviour is reduced only to the leading Hermite term of nonlinear functionals. Recently, Olenko and Omari (2019) proved the reduction principle for functionals of strongly dependent vector random fields. Components of such vector fields can possess different long-range dependences.…”

On asymptotics of discretized functionals of long-range dependent functional data

“…Also, it would be important to obtain similar results for the case of functionals of vector data, see Olenko and Omari (2019).…”

On Statistical Properties of Nonlinear Functionals of Random Fields

“…All theoretical results are supported by numerical studies. The main results of the thesis have been published in [4,9,10].…”