Probabilities of high excursions of Gaussian fields

Abstract

Let {ξ(t), t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with Eξ(t) ≡ 0, Dξ(t) ≡ 1, and continuous trajectories defined on the m-dimensional interval \( T \subset {\mathbb{R}^m} \). The paper is devoted to the problem of large excursions of the random field ξ. In particular, the asymptotic properties of the probability P = P{−v(t) < ξ(t) < u(t), t ∈ T}, when, for all t ∈ T, u(t), v(t) ⩾ χ, χ → ∞, are investigated. The work is a continuation of Rudzkis research started in [R. Rudzkis, Probabilities of large excursions of empirical processes and fields, Sov. Math., Dokl., 45(1):226–228, 1992]. It is shown that if the random field ξ satisfies certain smoothness and regularity conditions, then P = e−Q + Qo(1), where Q is a certain constructive functional depending on u, v, T, and the matrix function R(t) = cov(ξ′(t), ξ′(t)).