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Invariance principles for self-similar set-indexed random fields


Authors: Hermine Biermé and Olivier Durieu
Journal: Trans. Amer. Math. Soc. 366 (2014), 5963-5989
MSC (2010): Primary 60F17, 60G60, 60G18, 60G10, 60D05
DOI: https://doi.org/10.1090/S0002-9947-2014-06135-7
Published electronically: July 1, 2014
MathSciNet review: 3256190
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Abstract: For a stationary random field $ (X_j)_{j\in \mathbb{Z}^d}$ and some measure $ \mu $ on $ \mathbb{R}^d$, we consider the set-indexed weighted sum process

$\displaystyle S_n(A)=\sum _{j\in \mathbb{Z}^d}\mu (nA\cap R_j)^\frac 12 X_j,$

where $ R_j$ is the unit cube with lower corner $ j$. We establish a general invariance principle under a $ p$-stability assumption on the $ X_j$'s and an entropy condition on the class of sets $ A$. The limit processes are self-similar set-indexed Gaussian processes with continuous sample paths. Using Chentsov's type representations to choose appropriate measures $ \mu $ and particular sets $ A$, we show that these limits can be Lévy (fractional) Brownian fields or (fractional) Brownian sheets.

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Additional Information

Hermine Biermé
Affiliation: MAP5, UMR-CNRS 8145, Université Paris Descartes, PRES Sorbonne Paris Cité, 45 rue des Saints-Pères, 75006 Paris, France – and – Laboratoire de Mathématiques et Physique Théorique UMR-CNRS 7350, Fédération Denis Poisson FR-CNRS 2964, Université François-Rabelais de Tours, Parc de Grandmont, 37200 Tours, France
Email: hermine.bierme@mi.parisdescartes.fr

Olivier Durieu
Affiliation: Laboratoire de Mathématiques et Physique Théorique UMR-CNRS 7350, Fédération Denis Poisson FR-CNRS 2964, Université François-Rabelais de Tours, Parc de Grandmont, 37200 Tours, France
Email: olivier.durieu@lmpt.univ-tours.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-06135-7
Keywords: Dependent random field, invariance principle, set-indexed process, L\'evy fractional Brownian field, Chentsov's type representation, physical dependence measure, Vapnik-Chervonenkis dimension
Received by editor(s): September 10, 2012
Received by editor(s) in revised form: January 28, 2013
Published electronically: July 1, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.