University of Washington - Department of Statistics
We obtain an integral formula for the density of the maximum of smooth Gaussian processes. This expression induces explicit non-asymptotic lower and upper bounds which are in general asymptotic to the density. Moreover, these bounds allow us to derive simple asymptotic formulae 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 (1993) in a broader context, producing the rate of approximation for suitable variants of their asymptotic formulae. Our constructive approach relies on a geometric representation of Gaussian processes involving a unit speed parameterized curve embedded in the unit sphere.
This is joint work with Jean Diebolt, CNRS, Grenoble.