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.
Highlights
In the last decades a lot of papers have been dedicated to the sum of an infinite number of Poisson sources
It is proved that FH may be written as an integral with respect to a Poisson random measure and FH is called fractional Poisson field
The reader should be aware that the fractional Poisson field (fPf) we are dealing with has no relation -except the name- with the fractional Poisson process introduced in [4] for instance as a 1D Poisson process in a random time
Summary
In the last decades a lot of papers have been dedicated to the sum of an infinite number of Poisson sources. It is proved that FH may be written as an integral with respect to a Poisson random measure and FH is called fractional Poisson field (fPf). At this point, the reader should be aware that the fPf we are dealing with has no relation -except the name- with the fractional Poisson process introduced in [4] for instance as a 1D Poisson process in a random time. We concentrate on the finite-dimensional distributions of the fPf and on its moments From this point of view, there are obvious differences between fPf and fBf. We exhibit a representation of FH similar to the Chentsov one (see [16], Chapter 8). We will write VD for |B(0, 1)|, the volume of the unit Euclidean ball in RD, and SD−1 for the unit sphere in RD
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