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July 2009 Gaussian processes, kinematic formulae and Poincaré’s limit
Jonathan E. Taylor, Robert J. Adler
Ann. Probab. 37(4): 1459-1482 (July 2009). DOI: 10.1214/08-AOP439


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.


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Jonathan E. Taylor. Robert J. Adler. "Gaussian processes, kinematic formulae and Poincaré’s limit." Ann. Probab. 37 (4) 1459 - 1482, July 2009.


Published: July 2009
First available in Project Euclid: 21 July 2009

zbMATH: 1172.60006
MathSciNet: MR2546751
Digital Object Identifier: 10.1214/08-AOP439

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

Keywords: Euler characteristic , Excursion sets , Gaussian fields , geometry , intrinsic volumes , kinematic formulae , Poincaré’s limit

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 4 • July 2009
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