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Self-similar Random Fields and Rescaled Random Balls Models

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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.

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Authors and Affiliations

  1. Laboratory MAP5, Université Paris Descartes, CNRS UMR 8145, 45 rue des Saints-Pères, 75006, Paris, France

    Hermine Biermé & Anne Estrade

  2. Department of Mathematics, Uppsala University, P.O. Box 480, 751 06, Uppsala, Sweden

    Ingemar Kaj

Authors
  1. Hermine Biermé
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  2. Anne Estrade
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  3. Ingemar Kaj
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Correspondence to Hermine Biermé.

Additional information

This work was supported by ANR grant “mipomodim” ANR-05-BLAN-0017.

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Biermé, H., Estrade, A. & Kaj, I. Self-similar Random Fields and Rescaled Random Balls Models. J Theor Probab 23, 1110–1141 (2010). https://doi.org/10.1007/s10959-009-0259-x

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  • DOI: https://doi.org/10.1007/s10959-009-0259-x

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