A Gaussian kinematic formula

• 成果类型: Article
• 署名作者: Taylor, JE
• 署名单位: Stanford University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117905000000594
• 发表日期: 2006
• 页码: 122-158
• 关键词: excursion sets; random-fields; wiener space; MANIFOLDS; geometry; maxima; tube

摘要

In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl-Steiner tube formulae and the Chern-Federer kinematic fundamental formula. The probabilistic building blocks are smooth, real-valued random fields built up from i.i.d. copies of centered, unit-variance smooth Gaussian fields on a manifold M. Specifically, we consider random fields of the form f(p) = F(y(1) (p),..., y(k) (p)) for F is an element of C-2 (R-k ; R) and (y(1),..., y(k)) a vector of C-2 i.i.d. centered, unit-variance Gaussian fields. The analogue of the Weyl-Steiner formula for such Gaussian-related fields involves a power series expansion for the Gaussian, rather than Lebesgue, volume of tubes: that is, power series expansions related to the marginal distribution of the field f. The formal expansions of the Gaussian volume of a tube are of independent geometric interest. As in the classical Weyl-Steiner formulae, the coefficients in these expansions show up in a kinematic formula for the expected Euler characteristic, X, of the excursion sets M boolean AND f(-1) [u, +infinity) = M boolean AND y(-1) (F-1 [u, + infinity)) of the field f. The motivation for studying the expected Euler characteristic comes from the well-known approximation P[sup(p is an element of M) f(p) >= u] similar or equal to E[chi (f(-1) [u, +infinity))].