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January 2006 A Gaussian kinematic formula
Jonathan E. Taylor
Ann. Probab. 34(1): 122-158 (January 2006). DOI: 10.1214/009117905000000594

Abstract

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 fp=F(y1(p),…,yk(p)) for FC2(ℝk;ℝ) and (y1,…,yk) a vector of C2 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, χ, of the excursion sets Mf−1[u,+∞)=My−1(F−1[u,+∞)) of the field f.

The motivation for studying the expected Euler characteristic comes from the well-known approximation $\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$.

Citation

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Jonathan E. Taylor. "A Gaussian kinematic formula." Ann. Probab. 34 (1) 122 - 158, January 2006. https://doi.org/10.1214/009117905000000594

Information

Published: January 2006
First available in Project Euclid: 17 February 2006

zbMATH: 1094.60025
MathSciNet: MR2206344
Digital Object Identifier: 10.1214/009117905000000594

Subjects:
Primary: 53A17, 58A05, 60G15, 60G60
Secondary: 60G17, 60G70, 62M40

Rights: Copyright © 2006 Institute of Mathematical Statistics

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Vol.34 • No. 1 • January 2006
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