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.
Publié le : 1976-02-14
Classification:
Level crossings,
random fields,
normal bodies,
excursion sets,
characteristic of a normal body,
homogeneous Gaussian process,
mean values,
60G10,
60G15,
60G17,
53C65
@article{1176996176,
author = {Adler, Robert J. and Hasofer, A. M.},
title = {Level Crossings for Random Fields},
journal = {Ann. Probab.},
volume = {4},
number = {6},
year = {1976},
pages = { 1-12},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996176}
}
Adler, Robert J.; Hasofer, A. M. Level Crossings for Random Fields. Ann. Probab., Tome 4 (1976) no. 6, pp. 1-12. http://gdmltest.u-ga.fr/item/1176996176/