Volume of Tubes and Distribution of the Maxima of Gaussian Random Fields
نویسندگان
چکیده
1.1. Gaussian random fields on manifolds. Let X(p), p ∈ M̃ , be a realvalued random field on an orientable manifold M̃ . In this article, we consider a Gaussian random field with a smooth sample path and one-dimensional standard normal marginals (i.e., X(p) ∼ N(0, 1) for each p ∈ M̃). Let r(p, q) = Cov(X(p), X(q)) denote the covariance function. If r(p, q) is sufficiently smooth in a neighborhood of p = q for each p, then the sample path is smooth with probability one and we can differentiate X(p) to obtain Gaussian random fields ∇X, ∇X. We do not consider nonsmooth fields, such as Brownian motions or Ornstein-Uhlenbeck processes. Let t = (t), s = (s) denote local coordinate systems around p and q, respectively, and let Xi(p) = ∂X(p)/∂t. A metric
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