Poisson sphere counting processes with random radii
ESAIM: Probability and Statistics, Tome 20 (2016), pp. 417-431.

We consider a random sphere covering model made of random balls with interacting random radii of the product form R(r,ω)=rG(ω), based on a Poisson random measure ω(dy,dr) on R d ×R + . We provide sufficient conditions under which the corresponding random ball counting processes are well-defined, and we study the fractional behavior of the associated random fields. The main results rely on moment formulas for Poisson stochastic integrals with random integrands.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2016021
Classification : 60G60, 60F05
Mots-clés : Random balls, sphere counting, fractional processes, random fields, Poisson stochastic integrals, moment identities
Privault, Nicolas 1

1 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, 637371 Nanyang, Singapore.
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     url = {http://archive.numdam.org/articles/10.1051/ps/2016021/}
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Privault, Nicolas. Poisson sphere counting processes with random radii. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 417-431. doi : 10.1051/ps/2016021. http://archive.numdam.org/articles/10.1051/ps/2016021/

B. Bassan and E. Bona, Moments of stochastic processes governed by Poisson random measures. Comment. Math. Univ. Carolin. 31 (1990) 337–343. | Zbl

H. Biermé, Y. Demichel and A. Estrade, Fractional Poisson field on a finite set. Preprint (2011). | HAL

H. Biermé and A. Desolneux, Crossings of smooth shot noise processes. Ann. Appl. Probab. 22 (2012) 2240–2281. | DOI | Zbl

H. Biermé, A. Estrade and I. Kaj, Self-similar random fields and rescaled random balls models. J. Theoret. Probab. 23 (2010) 1110–1141. | DOI | Zbl

J.-C. Breton and C. Dombry, Rescaled weighted random ball models and stable self-similar random fields. Stochastic Process. Appl. 119 (2009) 3633–3652. | DOI | Zbl

Y. Demichel, Piling of multiscale random models. Preprint (2010). | HAL

I. Flint, X. Lu, N. Privault, D. Niyato and P. Wang, Performance analysis of ambient RF energy harvesting: A stochastic geometry approach. In IEEE Global Commun. Conf. GLOBECOM (2014) 1448–1453.

R. Gobard, Random balls model with dependence. J. Math. Anal. Appl. 423 (2015) 1284–1310. | DOI | Zbl

H.-B. Kong, I. Flint, D. Niyato and N. Privault, On the performance of wireless energy harvesting networks in a Boolean-Poisson model. In IEEE Green Commun. Syst. Networks Conf. ICC 2016 (2016).

J. Mecke, Stationäre zufällige Masse auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitstheorie Verw. Geb. 9 (1967) 36–58. | DOI | Zbl

N. Privault, Moments of Poisson stochastic integrals with random integrands. Probab. Math. Stat. 32 (2012) 227–239. | Zbl

N. Privault, Combinatorics of Poisson stochastic integrals with random integrands. In Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itô Chaos Expansions and Stochastic Geometry. Vol. 7 of Bocconi & Springer Series, edited by G. Peccati and M. Reitzner. Springer, Berlin (2016) 37–80.

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