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Lognormal \({\star}\) -scale invariant random measures
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  • Published: 17 January 2012

Lognormal \({\star}\) -scale invariant random measures

  • Romain Allez1,
  • Rémi Rhodes1 &
  • Vincent Vargas1 

Probability Theory and Related Fields volume 155, pages 751–788 (2013)Cite this article

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Abstract

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by Kahane in 1985 (or possibly extensions of this theory). As a by product, we also obtain an explicit characterization of the covariance structure of these measures. We also prove that qualitative properties such as long-range independence or isotropy can be read off the equation.

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Authors and Affiliations

  1. Université Paris-Dauphine, Ceremade, UMR 7534, Place du marchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Romain Allez, Rémi Rhodes & Vincent Vargas

Authors
  1. Romain Allez
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  2. Rémi Rhodes
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  3. Vincent Vargas
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Corresponding author

Correspondence to Rémi Rhodes.

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Cite this article

Allez, R., Rhodes, R. & Vargas, V. Lognormal \({\star}\) -scale invariant random measures. Probab. Theory Relat. Fields 155, 751–788 (2013). https://doi.org/10.1007/s00440-012-0412-9

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  • Received: 10 February 2011

  • Revised: 01 December 2011

  • Published: 17 January 2012

  • Issue Date: April 2013

  • DOI: https://doi.org/10.1007/s00440-012-0412-9

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Keywords

  • Random measure
  • Star equation
  • Scale invariance
  • Multiplicative chaos
  • Multifractal processes
  • Gaussian processes

Mathematics Subject Classification (2000)

  • 60G57
  • 60H25
  • 60G15
  • 60G18
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