Vol. 103, No. 1, 1982

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The expected measure of the level sets of a regular stationary Gaussian process

Salomon Benzaquen and Enrique M. Cabaña

Vol. 103 (1982), No. 1, 9–16
Abstract

If X(t1,t2,,td) is a sufficiently regular, centered, stationary Gaussian process, the (random) level set over a measurable domain T Rd

A(u) = {t ∈ T : X (t) = u}

is a d 1-dimensional manifold embedded in Rd. Our main result states that its expected measure is given by

                              2  √ ---
E μd−1(A (u)) = λ(T)E∥gradX ∥e−u ∕2∕  2π
(1)

where μd1(A) is the d 1-dimensional volume of the hypersurface A, λ is the Lebesgue measure on Rd and the variance of X is assumed to be one.

The expression (1) holds even for d = 1. In that case μ0(A) is a counting measure that gives the number of points in A. (μ1 and μ2 give respectively length and area.)

Mathematical Subject Classification 2000
Primary: 60G60
Secondary: 60G17
Milestones
Received: 18 September 1980
Revised: 2 June 1981
Published: 1 November 1982
Authors
Salomon Benzaquen
Enrique M. Cabaña