Probability distributions of extremes of self-similar Gaussian random fields

Abstract

We have obtained some upper bounds for the probability distribution of extremes of a selfsimilar Gaussian random field with stationary rectangular increments that are defined on the compact spaces. The probability distributions of extremes for the normalized self-similar Gaussian random fields with stationary rectangular increments defined in R 2 + have been presented. In our work we have used the techniques developed for the self-similar fields and based on the classical series analysis of the maximal probability bounding from below for the Gaussian fields.

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

“…Note that they used the estimate for N d 3 (ε), which coincides for H 1 = H 2 = 1 and a 1 = a 2 = 1 with the estimate for N d 2 (ε), and its form is very convenient to treat the case of self-similar random fields but is rather complicated for derivations for the general case. Our result for the case of metric d 3 appears in the form which is simpler but different from the corresponding bound in Kozachenko and Makogin (2014), first due to different estimate for N d 3 , but also since it is derived from Theorem 1.2, and derivations in Kozachenko and Makogin (2014) are based on another result. We postpone for further research the comparison of these bounds, in particular, by simulation studies.…”

“…The estimates for N d 1 and N d 2 can be found, for example, in Buldygin and Kozachenko (2000) and Kozachenko and Makogin (2014) correspondingly.…”

“…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 bounds for the distribution of suprema for self-similar Gaussian random fields were stated in Kozachenko and Makogin (2014) using the entropy approach and bounds on the increments in terms of the metrics d 3 . Note that they used the estimate for N d 3 (ε), which coincides for H 1 = H 2 = 1 and a 1 = a 2 = 1 with the estimate for N d 2 (ε), and its form is very convenient to treat the case of self-similar random fields but is rather complicated for derivations for the general case.…”

“…Remark 3.1. Note that in Kozachenko and Makogin (2014) several results were obtained for the rate of growth for self-similar Gaussian random fields using the bounds on the increments in terms of metrics d 3 . We state a more general result suitable for wider class of fields.…”

“…It can be shown by using the Minkowski inequality that E X(t 1 , t 2 ) − X(s 1 , s 2 ) Kozachenko and Makogin (2014), Lemma 2.4). Consider the field X(t 1 , t 2 ) over the rectangle T = [0, T 1 ] × [0, T 2 ].…”