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On long-range dependence of random measures

Part of: General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  11 January 2017

Daniel Vašata
Daniel Vašata*
Affiliation:
Czech Technical University in Prague
*
* Postal address: Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, Prague, CZ-160 00, Czech Republic. Email address: daniel.vasata@fit.cvut.cz
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Abstract

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This paper deals with long-range dependence of random measures on ℝd. By examples, it is demonstrated that one must be careful in order to define it consistently. Therefore, we define long-range dependence by a rather specific second-order condition and provide an equivalent formulation involving the asymptotic behaviour of the Bartlett spectrum near the origin. Then it is shown that the defining condition may be formulated less strictly when the additional isotropy assumption holds. Finally, we present an example of a long-range dependent random measure based on the 0-level excursion set of a Gaussian random field for which the corresponding spectral density and its asymptotics are explicitly derived.

Keywords

Long-range dependencerandom measureBartlett spectrum

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A40: Inequalities and extremum problems
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1235 - 1255
Copyright
Copyright © Applied Probability Trust 2017 

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