The mean Euler characteristic and excursion probability of Gaussian random fields with stationary increments

Abstract

Let X = {X(t), t ∈ R N } be a centered Gaussian random field with stationary increments and X(0) = 0. For any compact rectangle T ⊂ R N and u ∈ R, denote by Au = {t ∈ T : X(t) ≥ u} the excursion set. Under X(·) ∈ C 2 (R N ) and certain regularity conditions, the mean Euler characteristic of Au, denoted by E{ϕ(Au)}, is derived. By applying the Rice method, it is shown that, as u → ∞, the excursion probability P{sup t∈T X(t) ≥ u} can be approximated by E{ϕ(Au)} such that the error is exponentially smaller than E{ϕ(Au)}. This verifies the expected Euler characteristic heuristic for a large class of Gaussian random fields with stationary increments.1. Introduction. Let X = {X(t), t ∈ T } be a real-valued Gaussian random field on probability space (Ω, F, P), where T is the parameter set. The study of the excursion probability P{sup t∈T X(t) ≥ u} is a classical but very important problem in probability theory and has many applications in statistics and related areas. Many authors have developed various methods for precise approximations of P{sup t∈T X(t) ≥ u}. These include the double sum method [Piterbarg (1996a)], the tube method [Sun (1993)], the Euler characteristic method [Adler

Tail asymptotic behavior of the supremum of a class of chi-square processes

“…Then, by (19) and the Borell-TIS inequality for Gaussian random fields (cf. [12][Theorem 2.1.1]) we conclude that (17)…”

“…Numerous contributions have been devoted to the study of the tail asymptotics of the supremum of chi-square processes over compact intervals T ; see, e.g., [6,9,10,11] and the references therein, where the technique used is to transform the supremum of chi-square process into the supremum of a special Gaussian random field. We refer to, e.g., [12,13,14,15,16,17,18] for more discussions on the tail asymptotics (or excursion probability) of Gaussian and related fields.…”

Extremes of Gaussian random fields with regularly varying dependence structure

“…When α > β, then (7) holds with 1 instead of P b/a α ; see also Theorem 2.1 in [10] for the case T = ∞. We note in passing that in fact the Hölder continuity (5) is not needed to derive the asymptotics of (2), which will be shown later in our main theorems; necessary and sufficient conditions that guarantee the global Hölder continuity of X are presented in the deep contribution [1].…”

A note on global suprema of band-limited spherical random functions

“…Spherical random fields have recently drawn a lot of applied interest, especially in an astrophysical environment (see [6], [14]); closed form expressions for the density of their maxima and for excursion probabilities have been given in ( [10], [9], [16]). In particular, the latter references exploit the Gaussian Kinematic Fundamental formula by Adler and Taylor (see [1]) to approximate excursion probabilities by means of the expected value of the Euler-Poincarè characteristic for excursion sets.…”

“…denotes the derivative of the covariance function at the origin, see again ( [10], [9], [16]). When working on compact domains as the sphere, it is often of great interest to focus on sequences of band-limited random fields; for instance, a very powerful tool for data analysis is provided by fields which can be viewed as a sequence of wavelet transforms (at increasing frequencies) of a given isotropic spherical field T. More precisely, take b(.)…”