 
Summary: The Annals of Probability
2009, Vol. 37, No. 4, 14591482
DOI: 10.1214/08AOP439
© Institute of Mathematical Statistics, 2009
GAUSSIAN PROCESSES, KINEMATIC FORMULAE AND
POINCARÉ'S LIMIT
BY JONATHAN E. TAYLOR1,2 AND ROBERT J. ADLER1
Stanford University and Technion
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 LipschitzKilling
curvatures of these sets. Whereas our motivation is primarily probabilistic,
with statistical applications in the background, this formula has also an inter
pretation 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 approxi
mation to the canonical Gaussian process on the nsphere. The n limit,
which gives the final result, is handled via recent extensions of the classic
Poincaré limit theorem.
1. Introduction. The central aim of this paper is to describe a new result in
