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1 results

  • Excursions above a fixed level by n-dimensional random fields

  • Robert J. Adler
  • Journal:
    Journal of Applied Probability / Volume 13 / Issue 2 / June 1976
    Published online by Cambridge University Press:
    14 July 2016, pp. 276-289
    Print publication:
    June 1976

  • 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.