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May 2002 On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains
Akimichi Takemura, Satoshi Kuriki
Ann. Appl. Probab. 12(2): 768-796 (May 2002). DOI: 10.1214/aoap/1026915624

Abstract

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.

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Akimichi Takemura. Satoshi Kuriki. "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 (2) 768 - 796, May 2002. https://doi.org/10.1214/aoap/1026915624

Information

Published: May 2002
First available in Project Euclid: 17 July 2002

zbMATH: 1016.60042
MathSciNet: MR1910648
Digital Object Identifier: 10.1214/aoap/1026915624

Subjects:
Primary: 60G60
Secondary: 53C65

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.12 • No. 2 • May 2002
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