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July 1996 On the density of the maximum of smooth Gaussian processes
Jean Diebolt, Christian Posse
Ann. Probab. 24(3): 1104-1129 (July 1996). DOI: 10.1214/aop/1065725176


We obtain an integral formula for the density of the maximum of smooth Gaussian processes. This expression induces explicit nonasymptotic lower and upper bounds which are in general asymptotic to the density. Moreover, these bounds allow us to derive simple asymptotic formulas for the density with rate of approximation as well as accurate asymptotic bounds. In particular, in the case of stationary processes, the latter upper bound improves the well-known bound based on Rice's formula. In the case of processes with variance admitting a finite number of maxima, we refine recent results obtained by Konstant and Piterbarg in a broader context, producing the rate of approximation for suitable variants of their asymptotic formulas. Our constructive approach relies on a geometric representation of Gaussian processes involving a unit speed parameterized curve embedded in the unit sphere.


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Jean Diebolt. Christian Posse. "On the density of the maximum of smooth Gaussian processes." Ann. Probab. 24 (3) 1104 - 1129, July 1996.


Published: July 1996
First available in Project Euclid: 9 October 2003

zbMATH: 0863.60037
MathSciNet: MR1411489
Digital Object Identifier: 10.1214/aop/1065725176

Primary: 60G15 , 60G70
Secondary: 60G17

Keywords: Density , Differential geometry , extreme value , Gaussian processes , nonasymptotic formulas

Rights: Copyright © 1996 Institute of Mathematical Statistics


Vol.24 • No. 3 • July 1996
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