Summary: The Annals of Applied Probability
2005, Vol. 15, No. 1A, 254278
© Institute of Mathematical Statistics, 2005
ON THE DISTRIBUTION OF THE MAXIMUM OF A GAUSSIAN
FIELD WITH d PARAMETERS1
BY JEAN-MARC AZAÏS AND MARIO WSCHEBOR
Université Paul Sabatier and Universidad de la República
Let I be a compact d-dimensional manifold, let X :I R be a
Gaussian process with regular paths and let FI (u), u R, be the probability
distribution function of suptI X(t).
We prove that under certain regularity and nondegeneracy conditions,
FI is a C1-function and satisfies a certain implicit equation that permits
to give bounds for its values and to compute its asymptotic behavior as
u +. This is a partial extension of previous results by the authors in
the case d = 1.
Our methods use strongly the so-called Rice formulae for the moments of
the number of roots of an equation of the form Z(t) = x, where Z :I Rd
is a random field and x is a fixed point in Rd. We also give proofs for this kind
of formulae, which have their own interest beyond the present application.