Annals of Applied Probability

On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains

Satoshi Kuriki and Akimichi Takemura

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Consider a Gaussian random field with a finite Karhunen--Loève expansion of the form $Z(u) = \sum_{i=1}^n u_i z_i$, where $z_i$, $i=1,\ldots,n,$ are independent standard normal variables and $u=(u_1,\ldots,u_n)'$ ranges over an index set $M$, which is a subset of the unit sphere $S^{n-1}$ in $R^n$. Under a very general assumption that $M$ is a manifold with a piecewise smooth boundary, we prove the validity and the equivalence of two currently available methods for obtaining the asymptotic expansion of the tail probability of the maximum of $Z(u)$. One is the tube method, where the volume of the tube around the index set $M$ is evaluated. The other is the Euler characteristic method, where the expectation for the Euler characteristic of the excursion set is evaluated. General discussion on this equivalence was given in a recent paper by R. J. Adler. In order to show the equivalence we prove a version of the Morse theorem for a manifold with a piecewise smooth boundary.

Article information

Ann. Appl. Probab., Volume 12, Number 2 (2002), 768-796.

First available in Project Euclid: 17 July 2002

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Zentralblatt MATH identifier

Primary: 60G60: Random fields
Secondary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]

Chi-bar-squared distribution Gauss-Bonnet theorem Karhunen-Loève expansion kinematic fundamental formula Morse function Naiman's inequality


Takemura, Akimichi; Kuriki, Satoshi. On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains. Ann. Appl. Probab. 12 (2002), no. 2, 768--796. doi:10.1214/aoap/1026915624.

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