The Annals of Probability

Gaussian processes, kinematic formulae and Poincaré’s limit

Jonathan E. Taylor and Robert J. Adler

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We consider vector valued, unit variance Gaussian processes defined over stratified manifolds and the geometry of their excursion sets. In particular, we develop an explicit formula for the expectation of all the Lipschitz–Killing curvatures of these sets. Whereas our motivation is primarily probabilistic, with statistical applications in the background, this formula has also an interpretation as a version of the classic kinematic fundamental formula of integral geometry. All of these aspects are developed in the paper.

Particularly novel is the method of proof, which is based on a an approximation to the canonical Gaussian process on the n-sphere. The n→∞ limit, which gives the final result, is handled via recent extensions of the classic Poincaré limit theorem.

Article information

Ann. Probab., Volume 37, Number 4 (2009), 1459-1482.

First available in Project Euclid: 21 July 2009

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Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G60: Random fields 53A17: Kinematics 58A05: Differentiable manifolds, foundations
Secondary: 60G17: Sample path properties 62M40: Random fields; image analysis 60G70: Extreme value theory; extremal processes

Gaussian fields kinematic formulae excursion sets Poincaré’s limit Euler characteristic intrinsic volumes geometry


Taylor, Jonathan E.; Adler, Robert J. Gaussian processes, kinematic formulae and Poincaré’s limit. Ann. Probab. 37 (2009), no. 4, 1459--1482. doi:10.1214/08-AOP439.

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