Excursions above a fixed level by n-dimensional random fields
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For an n-dimensional random field X(t) we define the excursion set A of X(t) by A = [t ∊ S: X(t) ≧ u] for real u and compact S ⊂ Rn. We obtain a generalisation of the number of upcrossings of a one-dimensional stochastic process to random fields via a characteristic of the set A related to the Euler characteristic of differential topology. When X(t) is a homogeneous Gaussian field satisfying certain regularity conditions we obtain an explicit formula for the mean value of this characteristic.
1976 ◽
Vol 13
(02)
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pp. 276-289
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2017 ◽
Vol 54
(3)
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pp. 833-851
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2005 ◽
Vol 37
(01)
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pp. 108-133
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2014 ◽
Vol 51
(01)
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pp. 152-161
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1976 ◽
Vol 13
(02)
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pp. 377-379
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