Investigation of sample paths properties for some classes of φ-sub-Gaussian stochastic processes

Abstract

This paper investigates sample paths properties of ϕ-sub-Gaussian processes by means of entropy methods. Basing on a particular entropy integral, we treat the questions on continuity and the rate of growth of sample paths. The obtained results are then used to investigate the sample paths properties for a particular class of ϕ-sub-Gaussian processes related to the random heat equation. We derive the estimates for the distribution of suprema of such processes and evaluate their rate of growth.

Bounds for the Tail Distributions of Suprema of Sub-Gaussian Type Random Fields

“…, where σ k are introduced in the next theorem, θ = inf k γ k ε k . Theorem 3.1 below is an extension of the result stated in Sakhno and Vasylyk (2021)(Theorem 1), see also Hopkalo and Sakhno (2021)(Theorem 4). The proof presented in Sakhno and Vasylyk (2021)(Theorem 1) for the case of metrics d = d 1 works for a general metrics d as well.…”

“…Remark 2.1. Note that the most studied in the literature is the case of metrics d 1 , see, for example, Kozachenko et al (2020), Hopkalo and Sakhno (2021), where under this metrics different bounds for the function ρ X (t, s) = τ ϕ (X(t)−X(s)) where considered and corresponding bounds for the distributions of suprema were stated. In Theorem 2.1 (following Theorem 5.4 in Sakhno (2023b)) the improved bound is presented in comparison with the analogous results stated in Kozachenko et al (2020) (Corollary 3.1) and Hopkalo and Sakhno (2021) (Corollary 2).…”

“…The main theory for the spaces of ϕ-sub-Gaussian random variables and stochastic processes was presented in Buldygin and Kozachenko (2000); Giuliano et al (2003); Kozachenko and Ostrovskij (1986) followed by numerous further studies. Various classes of ϕ-sub-Gaussian processes and fields were studied, in particular, in Beghin, Kozachenko, Orsingher, and Sakhno (2007); Hopkalo and Sakhno (2021); Kozachenko and Olenko (2016); Vasylyk (2018, 2020).…”

“…Theorem 1.2 with a particular metric space (T, d) has been applied in Hopkalo and Sakhno (2021); Kozachenko et al (2020); Sakhno (2023b) for studying solutions to partial differential equations with random initial condition, in Sakhno (2023a) for evaluation of suprema of spherical random fields, in Kozachenko and Olenko (2016) for developing uniform approximation schemes for ϕ-sub-Gaussian processes. This theorem allows to calculate the bounds for the distribution of suprema in the closed form.…”

“…which is more convenient for estimation of the expression (3.1) (needed to apply Theorem 3.1) than the estimate from Theorem 2.1. We refer for more details to Hopkalo and Sakhno (2021), Sakhno and Vasylyk (2021).…”