Fractional Poisson Fields

This paper considers random balls in a D-dimensional Euclidean space whose centers are prescribed by a homogeneous Poisson point process and whose radii are prescribed by a specific power law. A random field is constructed by counting the number of covering balls at each point. Even though it is not Gaussian, this field shares the same covariance function as the fractional Brownian field (fBf). By analogy it is called fractional Poisson field (fPf). In this paper, we are mainly interested in the simulation of fPfs with index H in (0,1/2) and in the estimation of the H index. Our method is based on the analysis of structure functions. The fPf exhibits a multifractal behavior, contrary to that of the fBf which is monofractal.

[1]  Ingemar Kaj,et al.  Self-similar Random Fields and Rescaled Random Balls Models , 2010 .

[2]  Pierre Chainais,et al.  Infinitely Divisible Cascades to Model the Statistics of Natural Images , 2007, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[3]  T. Mattfeldt.  Stochastic Geometry and Its Applications , 1996 .

[4]  Gabriel Lang,et al.  Quadratic variations and estimation of the local Hölder index of a gaussian process , 1997 .

[5]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[6]  B. Mandelbrot,et al.  A CLASS OF MICROPULSES AND ANTIPERSISTENT FRACTIONAL BROWNIAN MOTION , 1995 .

[7]  Claude Tricot.  Function norms and fractal dimension , 1997 .

[8]  L p -Norms and Fractal Dimensions of Continuous Function Graphs , 2010 .

[9]  D. Daley.  The definition of a multi-dimensional generalization of shot noise , 1971, Journal of Applied Probability.

[10]  M. Taqqu,et al.  Convergence to Fractional Brownian Motion and to the Telecom Process: the Integral Representation Approach , 2008 .

[11]  On the dimension of the boundary of clumps in a multi-type boolean model , 1996 .

[12]  Yann Demichel.  Analyse fractale d'une famille de fonctions aléatoires : les fonctions de bosses , 2006 .

[13]  V. Schmidt,et al.  NORMAL CONVERGENCE OF MULTIDIMENSIONAL SHOT NOISE AND RATES OF THIS CONVERGENCE , 1985 .

[14]  V. Schmidt,et al.  Scaling limits for random fields with long-range dependence , 2007, 0707.3729.

[15]  M. Taqqu,et al.  Small and Large Scale Behavior of the Poissonized Telecom Process , 2004 .

[16]  A. Estrade,et al.  Poisson random balls: self-similarity and X-ray images , 2006, Advances in Applied Probability.