Mathematics – Probability
Scientific paper
2006-02-24
Annals of Probability 2006, Vol. 34, No. 1, 122-158
Mathematics
Probability
Published at http://dx.doi.org/10.1214/009117905000000594 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins
Scientific paper
10.1214/009117905000000594
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 $F\in C^2(\mathbb{R}^k;\mathbb{R})$ 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, $\chi$, of the excursion sets $M\cap f^{-1}[u,+\infty)=M\cap y^{-1}(F^{-1}[u,+\infty))$ of the field $f$. The motivation for studying the expected Euler characteristic comes from the well-known approximation $\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$.
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