The Annals of Probability

On the perimeter of excursion sets of shot noise random fields

Hermine Biermé and Agnès Desolneux

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Abstract

In this paper, we use the framework of functions of bounded variation and the coarea formula to give an explicit computation for the expectation of the perimeter of excursion sets of shot noise random fields in dimension $n\geq1$. This will then allow us to derive the asymptotic behavior of these mean perimeters as the intensity of the underlying homogeneous Poisson point process goes to infinity. In particular, we show that two cases occur: we have a Gaussian asymptotic behavior when the kernel function of the shot noise has no jump part, whereas the asymptotic is non-Gaussian when there are jumps.

Article information

Source
Ann. Probab., Volume 44, Number 1 (2016), 521-543.

Dates
Received: October 2013
Revised: July 2014
First available in Project Euclid: 2 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1454423048

Digital Object Identifier
doi:10.1214/14-AOP980

Mathematical Reviews number (MathSciNet)
MR3457393

Zentralblatt MATH identifier
1343.60060

Subjects
Primary: 60G60: Random fields 60E10: Characteristic functions; other transforms 26B30: Absolutely continuous functions, functions of bounded variation 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]
Secondary: 60G10: Stationary processes 60F05: Central limit and other weak theorems 60E07: Infinitely divisible distributions; stable distributions

Keywords
Shot noise excursion set stationary process Poisson process characteristic function functions of bounded variation coarea formula

Citation

Biermé, Hermine; Desolneux, Agnès. On the perimeter of excursion sets of shot noise random fields. Ann. Probab. 44 (2016), no. 1, 521--543. doi:10.1214/14-AOP980. https://projecteuclid.org/euclid.aop/1454423048


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References

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