## Electronic Communications in Probability

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

#### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 18 (2013), paper no. 11, 13 pp.

Dates
Accepted: 7 February 2013
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465315550

Digital Object Identifier
doi:10.1214/ECP.v18-1939

Mathematical Reviews number (MathSciNet)
MR3033594

Zentralblatt MATH identifier
1319.60105

Rights

#### Citation

Biermé, Hermine; Demichel, Yann; Estrade, Anne. Fractional Poisson field and fractional Brownian field: why are they resembling but different?. Electron. Commun. Probab. 18 (2013), paper no. 11, 13 pp. doi:10.1214/ECP.v18-1939. https://projecteuclid.org/euclid.ecp/1465315550

#### References

• Bassan, Bruno; Bona, Elisabetta. Moments of stochastic processes governed by Poisson random measures. Comment. Math. Univ. Carolin. 31 (1990), no. 2, 337–343.
• Biermé, Hermine; Estrade, Anne; Kaj, Ingemar. Self-similar random fields and rescaled random balls models. J. Theoret. Probab. 23 (2010), no. 4, 1110–1141.
• Breton, Jean-Christophe; Dombry, Clément. Functional macroscopic behavior of weighted random ball model. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 177–196.
• Beghin, L.; Orsingher, E. Fractional Poisson processes and related planar random motions. Electron. J. Probab. 14 (2009), no. 61, 1790–1827.
• Chainais P., phInfinitely Divisible Cascades to Model the Statistics of Natural Images, IEEE Trans. Pattern analysis and Machine intelligence, 29, 12, 2105–2119, (2007).
• Cioczek-Georges, R.; Mandelbrot, B. B. A class of micropulses and antipersistent fractional Brownian motion. Stochastic Process. Appl. 60 (1995), no. 1, 1–18.
• Cohen, Serge; Taqqu, Murad S. Small and large scale behavior of the Poissonized Telecom process. Methodol. Comput. Appl. Probab. 6 (2004), no. 4, 363–379.
• Guyon, Xavier; LeÃ³n, José. Convergence en loi des $H$-variations d'un processus gaussien stationnaire sur ${\bf R}$. (French) [Convergence in law of the $H$-variations of a stationary Gaussian process in ${\bf R}$] Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 3, 265–282.
• Heinrich, Lothar; Schmidt, Volker. Normal convergence of multidimensional shot noise and rates of this convergence. Adv. in Appl. Probab. 17 (1985), no. 4, 709–730.
• Istas, Jacques; Lang, Gabriel. Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), no. 4, 407–436.
• Kaj, Ingemar; Leskelä, Lasse; Norros, Ilkka; Schmidt, Volker. Scaling limits for random fields with long-range dependence. Ann. Probab. 35 (2007), no. 2, 528–550.
• Kaj, Ingemar; Taqqu, Murad S. Convergence to fractional Brownian motion and to the Telecom process: the integral representation approach. In and out of equilibrium. 2, 383–427, Progr. Probab., 60, Birkhäuser, Basel, 2008.
• Peccati, Giovanni; Taqqu, Murad S. Central limit theorems for double Poisson integrals. Bernoulli 14 (2008), no. 3, 791–821.
• Perrin E., Harba R., Jennane R. and Iribarren I. phFast and exact synthesis for 1-D fractional Brownian motion and fractional Gaussian noises, Signal Processing Letters IEEE, Vol.9 (1), 382–384, (2002).
• Sato, Yumiko. Distributions of stable random fields of Chentsov type. Nagoya Math. J. 123 (1991), 119–139.
• Samorodnitsky, Gennady; Taqqu, Murad S. Stable non-Gaussian random processes. Stochastic models with infinite variance. Stochastic Modeling. Chapman &Hall, New York, 1994. xxii+632 pp. ISBN: 0-412-05171-0
• Stoyan, D.; Kendall, W. S.; Mecke, J. Stochastic geometry and its applications. With a foreword by D. G. Kendall. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. John Wiley & Sons, Ltd., Chichester, 1987. 345 pp. ISBN: 0-471-90519-4
• Takenaka, Shigeo. Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 (1991), 1–12.