Probabilities of high excursions of Gaussian fields.

Let { ξ( t) , t ∈ T} be a differentiable (in the mean-square sense) Gaussian random field with E ξ( t) ≡ 0, D ξ( 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 + 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)). [ABSTRACT FROM AUTHOR]