Sojourn measures of Student and Fisher–Snedecor random fields

Abstract

Limit theorems for the volumes of excursion sets of weakly and strongly dependent heavy-tailed random fields are proved. Some generalizations to sojourn measures above moving levels and for cross-correlated scenarios are presented. Special attention is paid to Student and Fisher-Snedecor random fields. Some simulation results are also presented. Sojourn measures of Student and Fisher-Snedecor random fields3 The paper has three aims. One is to provide explicit, albeit asymptotic, formulae for the distribution of the volume of excursion sets of a class of strongly dependent random fields. The second one is to derive asymptotic results for heavy-tailed random fields. Finally, the third aim is to generalize the previous findings to sojourn measures above moving levels and for cross-correlated scenarios.There is, therefore, a need for models that are able to display strongly dependent heavytailed behaviour and yet are sufficiently simple to allow analysis. To obtain explicit results we detail the underlying structure of random fields. Namely, a basic assumption of the analysis is that we examine functionals of vector Gaussian random fields, in particular, Student and Fisher-Snedecor random fields. Consult [3,15,16,47] on excursion sets of chi-square, Student and Fisher-Snedecor random fields and their importance for image analysis and studies of brain function. Other results on sojourn measures of chi-square random fields can be found in [23,27,29,30].Minkowski functionals are widely used to characterise geometric properties of random fields, in particular in the analysis of cosmic microwave background radiation, see [36,38]. In this paper we investigate the first Minkowski functional of random fields and its expansions into multidimensional Hermite polynomials, see some one-dimensional/discrete counterparts in [18,20]. To have a complete account of results on asymptotic distributions of sojourn measures for functions of vector random fields, we also prove corresponding theorems for weakly dependent scenarios.The remainder of the paper is structured as follows. In Sections 2-4, we introduce the necessary background from the theory of random fields and briefly review some definitions and notation on the first Minkowski functional, multidimensional Hermite expansions, and Student and Fisher-Snedecor random fields. We start Sections 5 and 7 with generalizations and corrections of some classical asymptotic results to arbitrary sets and vector fields. With this in hand, we continue Sections 5 and 7 by new results for the first Minkowski functional of Student and Fisher-Snedecor random fields. In Section 7, we also show how to lift these results to sojourn measures above moving levels and for cross-correlated underlying vector fields. Sections 6 and 8 provide the proofs of all theorems and lemmata in the article. Simulation results on the limit distributions of areas of excursion sets for two types of images are given in Section 9. Short conclusions are made in Section 10.In this paper, we only consider real-valued random f...

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

“…Theorem 3.1. (Leonenko and Olenko (2014)) Suppose that ξ (x) , x ∈ R n , satisfies Assumption 3.1 and HrankG(·) = κ ≥ 1. If a limit distribution exists for at least one of the random variables K r √ V arK r and K r,κ V arK r,κ , then the limit distribution of the other random variable also exists, and the limit distributions coincide when r → ∞.…”

“…Theorem 3.2. (Leonenko and Olenko (2014)) Let ξ (x) , x ∈ R n , be a homogeneous isotropic Gaussian random field. If Assumptions 3.1 and 3.2 hold, α ∈ 0, n/κ , then for r → ∞ the random variables…”

Limit theorems for filtered long-range dependent random fields

“…Theorem 4 [21] Suppose that ξ (x) , x ∈ R n , satisfies Assumption 1 and HrankS(·) = κ ≥ 1. If a limit distribution exists for at least one of the random variables K r √ V arK r and K r,κ V arK r,κ , then the limit distribution of the other random variable also exists, and the limit distributions coincide when r → ∞.…”

“…Theorem 5 [21] Let ξ (x) , x ∈ R n , be a homogeneous isotropic Gaussian random field. If Assumptions 1 and 2 hold, then for r → ∞ the random variables X r,κ (∆) := r κα/2−n L −κ/2 (r)…”

Reduction Principle for Functionals of Vector Random Fields

“…The first Minkowski functional and its Hermite expansions were discussed in the one-dimensional case for discrete-time processes in . Some recent developments in multidimensional and continuous counterparts can be found in Leonenko and Olenko 2014. Limit theorems are the central topic in the theory of probability. Considerable attention has been paid to study the asymptotic behaviour of sums (or integrals) of non-linear functionals of stationary Gaussian random processes and fields.…”