Piling of multiscale random models

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 forced to be in a finite interval with a power-law distribution. Random fields are constructed by counting the number of covering balls at each point. We are mainly interested in the simulation of these fields and in the empirical estimation of its index H. Finally simulations are given. Introduction Various random fields are obtained by summing elementary patterns properly rescaled and normalized. 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 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, 7]) 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 build elementary fields by counting the number of balls whose centers and radii are distributed according a Poisson point process. If the centers are uniformly distributed in the space, we force the radii to be in each given slice (αj+1, αj ] (α ∈ (0, 1) is fixed and j ranges in Z). Moreover their distribution has a power-law density of the type r−D−1+2H , H ∈ R. Next we build a piling field Fjmin,jmax summing all the slices from jmin to jmax. We are mainly interested in the simulation of fields obtained by piling elementary slices and in the estimation of its index H. Let us note that simulating such fields appears as very tractable since the basic objects are balls and the basic operation consists in counting. We first simulate each slice and then proceed to the piling of the slices. This procedure is similar to the construction of the ”general multitype Boolean model” in [10]. Concerning the index estimation, we use the structure functions as introduced in [17]. Roughly speaking, the q-structure function of a given function f is equal to the Lq-norm of the ε-increments of f . We use these tools to provide two different empirical estimators of the H index: The first one takes into account the small scales and the second one the intermediate scales. Outline of the paper The random fields we deal with are introduced in Section 1. First the piling field is obtained by summing the elementary slices between a lower and an upper scale. In Section 2, 1991 Mathematics Subject Classification. primary 60G60; secondary 60D05, 60G55, 62M40, 62F10.