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Fractional Poisson field and fractional Brownian field: why are they resembling but different?


 
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1. Title Title of document Fractional Poisson field and fractional Brownian field: why are they resembling but different?
 
2. Creator Author's name, affiliation, country Hermine Biermé; Université Paris Descartes; France
 
2. Creator Author's name, affiliation, country Yann Demichel; Université Paris Ouest Nanterre La Défense; France
 
2. Creator Author's name, affiliation, country Anne Estrade; Université Paris Descartes; France
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Random field; Poisson random measure; Poisson point process; Fractional Brownian field; Fractional process; Chentsov field; Hurst index
 
3. Subject Subject classification 60G60; 60D05; 60G55; 62M40; 62F10
 
4. Description Abstract The fractional Poisson field (fPf) is constructed by considering the number of balls falling down on each point of $\mathbb R^D$, when the centers and the radii of the balls are thrown at random following a Poisson point process in $\mathbb R^D\times \mathbb R^+$ with an appropriate intensity measure. It provides a simple description for a non Gaussian random field that is centered, has stationary increments and has the same covariance function as the fractional Brownian field (fBf). The present paper is concerned with specific properties of the fPf, comparing them to their analogues for the fBf. On the one hand, we concentrate on the finite-dimensional distributions which reveal strong differences between the Gaussian world of the fBf and the Poissonnian world of the fPf. We provide two different representations for the marginal distributions of the fPf: as a Chentsov field, and on a regular grid in $\mathbb R^D$ with a numerical procedure for simulations. On the other hand, we prove that the Hurst index estimator based on quadratic variations which is commonly used for the fBf is still strongly consistent for the fPf. However the computations for the proof are very different from the usual ones.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) MATAIM ANR-09-BLAN-0029-01
 
7. Date (YYYY-MM-DD) 2013-02-07
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1939
 
10. Identifier Digital Object Identifier 10.1214/ECP.v18-1939
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 18
 
12. Language English=en en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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