Self-similar Random Fields and Rescaled Random Balls Models

Hermine Biermé (), Anne Estrade () and Ingemar Kaj ()
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Hermine Biermé: Université Paris Descartes
Anne Estrade: Université Paris Descartes
Ingemar Kaj: Uppsala University

Journal of Theoretical Probability, 2010, vol. 23, issue 4, 1110-1141

Abstract: Abstract We study generalized random fields which arise as rescaling limits of spatial configurations of uniformly scattered random balls as the mean radius of the balls tends to 0 or infinity. Assuming that the radius distribution has a power-law behavior, we prove that the centered and renormalized random balls field admits a limit with self-similarity properties. Our main result states that all self-similar, translation- and rotation-invariant Gaussian fields can be obtained through a unified zooming procedure starting from a random balls model. This approach has to be understood as a microscopic description of macroscopic properties. Under specific assumptions, we also get a Poisson-type asymptotic field. In addition to investigating stationarity and self-similarity properties, we give L 2-representations of the asymptotic generalized random fields viewed as continuous random linear functionals.

Keywords: Self-similarity; Generalized random field; Poisson point process; Fractional Poisson field; Fractional Brownian field; 60G60; 60G78; 60G20; 60D05; 60G55; 60G10; 60F05 (search for similar items in EconPapers)
Date: 2010
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-009-0259-x

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