The Annals of Probability

Level Crossings for Random Fields

Robert J. Adler and A. M. Hasofer

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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.

Article information

Ann. Probab., Volume 4, Number 1 (1976), 1-12.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G10: Stationary processes
Secondary: 60G15: Gaussian processes 60G17: Sample path properties 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]

Level crossings random fields normal bodies excursion sets characteristic of a normal body homogeneous Gaussian process mean values


Adler, Robert J.; Hasofer, A. M. Level Crossings for Random Fields. Ann. Probab. 4 (1976), no. 1, 1--12. doi:10.1214/aop/1176996176.

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