This paper investigates solutions of hyperbolic diffusion equations in R 3 with random initial conditions. The solutions are given as spatial-temporal random fields. Their restrictions to the unit sphere S 2 are studied. All assumptions are formulated in terms of the angular power spectrum or the spectral measure of the random initial conditions. Approximations to the exact solutions are given. Upper bounds for the meansquare convergence rates of the approximation fields are obtained. The smoothness properties of the exact solution and its approximation are also investigated. It is demonstrated that the Hölder-type continuity of the solution depends on the decay of the angular power spectrum. Conditions on the spectral measure of initial conditions that guarantee short or long-range dependence of the solutions are given. Numerical studies are presented to verify the theoretical findings.Stochastic partial differential equations; Hyperbolic diffusion equation; Spherical random field; Hölder continuity; Long-range dependence; Approximation errors; Cosmic microwave background arXiv:1912.08378v1 [math.PR]

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.

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We prove the reduction principle for asymptotics of functionals of vector random fields with weakly and strongly dependent components. These functionals can be used to construct new classes of random fields with skewed and heavy-tailed distributions. Contrary to the case of scalar long-range dependent random fields, it is shown that the asymptotic behaviour of such functionals is not necessarily determined by the terms at their Hermite rank. The results are illustrated by an application to the first Minkowski functional of the Student random fields. Some simulation studies based on the theoretical findings are also presented.

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