Annals of Probability

On the density of the maximum of smooth Gaussian processes

Jean Diebolt and Christian Posse

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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.

Article information

Ann. Probab., Volume 24, Number 3 (1996), 1104-1129.

First available in Project Euclid: 9 October 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G70: Extreme value theory; extremal processes
Secondary: 60G17: Sample path properties

Differential geometry Gaussian processes extreme value nonasymptotic formulas density


Diebolt, Jean; Posse, Christian. On the density of the maximum of smooth Gaussian processes. Ann. Probab. 24 (1996), no. 3, 1104--1129. doi:10.1214/aop/1065725176.

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