Invariance principles for self-similar set-indexed random fields
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- by Hermine Biermé and Olivier Durieu PDF
- Trans. Amer. Math. Soc. 366 (2014), 5963-5989 Request permission
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 \[ S_n(A)=\sum _{j\in \mathbb {Z}^d}\mu (nA\cap R_j)^\frac 12 X_j,\] 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.References
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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
- Received by editor(s): September 10, 2012
- Received by editor(s) in revised form: January 28, 2013
- Published electronically: July 1, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3256190