Spectral theory for random closed sets and estimating the covariance via frequency space



Advances in Applied Probability

Spectral theory for random closed sets and estimating the covariance via frequency space

Karsten Koch, Joachim Ohser, and Katja Schladitz

Source: Adv. in Appl. Probab. Volume 35, Number 3 (2003), 603-613.

Abstract

A spectral theory for stationary random closed sets is developed and provided with a sound mathematical basis. The definition and a proof of the existence of the Bartlett spectrum of a stationary random closed set as well as the proof of a Wiener-Khinchin theorem for the power spectrum are used to two ends. First, well-known second-order characteristics like the covariance can be estimated faster than usual via frequency space. Second, the Bartlett spectrum and the power spectrum can be used as second-order characteristics in frequency space. Examples show that in some cases information about the random closed set is easier to obtain from these characteristics in frequency space than from their real-world counterparts.

Primary Subjects: 60D05
Secondary Subjects: 62G05, 42B10, 62M40
Keywords: Random set; Bartlett spectrum; fast Fourier transform; power spectrum

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aap/1059486820
Digital Object Identifier: doi:10.1239/aap/1059486820
Mathematical Reviews number (MathSciNet): MR1990606
Zentralblatt MATH identifier: 02072234

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