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February, 1976 Level Crossings for Random Fields
Robert J. Adler, A. M. Hasofer
Ann. Probab. 4(1): 1-12 (February, 1976). DOI: 10.1214/aop/1176996176


For an $n$-dimensional random field $X(\mathbf{t})$ we define the excursion set $A$ of $X(\mathbf{t})$ by $A = \{\mathbf{t} \in \mathbf{I}_0: X(\mathbf{t}) \geqq u\}$, where $I_0$ is the unit cube in $R^n.$ It is shown that the natural generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields is via the characteristic of the set $A$ introduced by Hadwiger (1959). This characteristic is related to the number of connected components of $A$. A formula is obtained for the mean value of this characteristic when $n = 2, 3$. This mean value is calculated explicitly when $X(\mathbf{t})$ is a homogeneous Gaussian field satisfying certain regularity conditions.


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Robert J. Adler. A. M. Hasofer. "Level Crossings for Random Fields." Ann. Probab. 4 (1) 1 - 12, February, 1976.


Published: February, 1976
First available in Project Euclid: 19 April 2007

zbMATH: 0329.60017
MathSciNet: MR405559
Digital Object Identifier: 10.1214/aop/1176996176

Primary: 60G10
Secondary: 53C65 , 60G15 , 60G17

Keywords: characteristic of a normal body , Excursion sets , homogeneous Gaussian process , level crossings , mean values , normal bodies , Random fields

Rights: Copyright © 1976 Institute of Mathematical Statistics


Vol.4 • No. 1 • February, 1976
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