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. Introduction The random fields under consideration in this paper have been introduced in [15] and [2] as a limit of a rescaled shot-noise. More precisely, random balls in a D-dimensional Euclidean space are considered: the centers are prescribed by a homogeneous Poisson point process in RD and the radii are prescribed by a power law. A shot-noise field is constructed by counting the number of covering balls (see [7, 12] for details on shot-noise processes) at each point. When there are enough balls with arbitrary small volumes, the associated field exhibits fractal properties at small scale and some global self-similar properties. In particular, its covariance function is a homogeneous function whose degree depends on the power exponent. Hence it shares the same covariance as a fractional Brownian field and it is called fractional Poisson field (fPf). For the fPf, the described procedure yields indices that range in (0, 1/2). A pioneer work in this area is due to Cioczek-Georges and Mandelbrot [5] where a sum of random micropulses in dimension one, or generalizations in higher dimensions, are properly rescaled and normalized in order to get a fractional Brownian field of index H < 1/2 (antipersistent fBf). In that paper, it is emphasized that the power law distribution prescribed for the length of the micropulses makes it impossible to get H > 1/2. Using similar models in dimension one, recent works ([6, 10, 16]) have examined the internet traffic modeling. The resulting signal is proved to exhibit a long range dependence (H > 1/2), in accordance with observations. Such a range for index H is made possible either by prescribing the connection lengths with heavy tails, or by forcing the number of long connections. In the present paper, we are mainly interested in the simulation of an fPf and the estimation of its index. Let us note that simulating an fPf appears as very tractable since the basic objects are balls and the basic operation consists in counting. Moreover, the possibility of changing balls into other templates or changing the homogeneous Poisson process of centers into another point process yields a very large choice of patterns (see [4] for instance). In order to get simulations which are rapidly accurate at all scales, we force random balls with prescribed radii in each given slice (αj+1, αj ] (α ∈ (0, 1) is fixed and j ranges in Z). We first 2000 Mathematics Subject Classification. primary 60G60; secondary 60D05, 60G55, 62M40, 62F10, 28A80.