Abstract
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−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)).
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The research is partially supported by the European Union Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania” within the framework of the Measure for Enhancing Mobility of Scholars and Other Researchers and the Promotion of Student Research (VP1-3.1-ŁMM-01) of the Program of Human Resources Development Action Plan.
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Rudzkis, R., Bakshaev, A. Probabilities of high excursions of Gaussian fields. Lith Math J 52, 196–213 (2012). https://doi.org/10.1007/s10986-012-9167-3
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DOI: https://doi.org/10.1007/s10986-012-9167-3