Advances in Applied Probability

On the capacity functional of excursion sets of Gaussian random fields on ℝ2

Marie Kratz and Werner Nagel

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Abstract

When a random field (Xt,t∈ℝ2) is thresholded on a given level u, the excursion set is given by its indicator 1[u, ∞)(Xt). The purpose of this work is to study functionals (as established in stochastic geometry) of these random excursion sets as, e.g. the capacity functional as well as the second moment measure of the boundary length. It extends results obtained for the one-dimensional case to the two-dimensional case, with tools borrowed from crossings theory, in particular, Rice methods, and from integral and stochastic geometry.

Article information

Source
Adv. in Appl. Probab., Volume 48, Number 3 (2016), 712-725.

Dates
First available in Project Euclid: 19 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aap/1474296311

Mathematical Reviews number (MathSciNet)
MR3568888

Zentralblatt MATH identifier
1351.60063

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G60: Random fields

Keywords
Capacity functional crossings excursion set Gaussian field growing circle method Rice formula second moment measure sweeping line method stereology stochastic geometry

Citation

Kratz, Marie; Nagel, Werner. On the capacity functional of excursion sets of Gaussian random fields on ℝ 2. Adv. in Appl. Probab. 48 (2016), no. 3, 712--725. https://projecteuclid.org/euclid.aap/1474296311


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References

  • Adler, R. J. (1981). The Geometry of Random Fields. John Wiley, Chichester.
  • Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.
  • Adler, R. J., Samorodnitsky, G. and Taylor, J. E. (2010). Excursion sets of three classes of stable random fields. Adv. Appl. Prob. 42, 293–318.
  • Azaïs, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. John Wiley, Hoboken, NJ.
  • Brodtkorb, P. A. (2006). Evaluating nearly singular multinormal expectations with application to wave distributions. Methodol. Comput. Appl. Prob. 8, 65–91.
  • Chiu, S. N., Stoyan, D., Kendall, W. S. and Mecke, J. (2013). Stochastic Geometry and its Applications, 3rd edn. John Wiley, Chichester.
  • Cramér, H. and Leadbetter, M. R. (1967). Stationary and Related Stochastic Processes. John Wiley, New York.
  • Estrade, A., Iribarren, I. and Kratz, M. (2011). Chord-length distribution functions and Rice formulae. Application to random media. Extremes 15, 333–352.
  • Kratz, M. (2006). Level crossings and other level functionals of stationary Gaussian processes. Prob. Surveys 3, 230–288.
  • Kratz, M. and León, J. R. (2001). Central limit theorems for level functionals of stationary Gaussian processes and fields. J. Theoret. Prob. 14, 639–672.
  • Lindgren, G. (1972). Wave-length and amplitude in Gaussian noise. Adv. Appl. Prob. 4, 81–108.
  • Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.
  • Mercadier, C. (2006). Numerical bounds for the distributions of the maximum of some one- and two-parameter Gaussian processes. Adv. Appl. Prob. 38, 149–170.
  • Molchanov, I. (2005). Theory of Random Sets. Springer, London.
  • Piterbarg, V. I. (1996). Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society, Providence, RI.
  • Preparata, F. P. and Shamos, M. I. (1985). Computational Geometry: An Introduction. Springer, New York.
  • Rychlik, I. (1987). Joint distribution of successive zero crossing distances for stationary Gaussian processes. J. Appl. Prob. 24, 378–385.
  • Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.
  • Serra, J. (1984). Image Analysis and Mathematical Morphology. Academic Press, London.
  • Weiss, V. and Nagel, W. (1994). Second-order stereology for planar fibre processes. Adv. Appl. Prob. 26, 906–918.
  • Wschebor, M. (1985). Surfaces Aléatoires (Lecture Notes Math. 1147). Springer, Berlin.