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

DOI: 10.2140/pjm.1982.103.9

Abstract

If X(t₁,t₂,⋯,t_d) is a sufficiently regular, centered, stationary Gaussian process, the (random) level set over a measurable domain T ⊂ R^d

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

(1)

where μ_d−1(A) is the d − 1-dimensional volume of the hypersurface A, λ is the Lebesgue measure on R^d 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