Shot-noise Fields Research Articles

Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields

Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields

On the finiteness of the moments of the measure of level sets of random fields

General conditions on smooth real valued random fields are given that ensure the finiteness of the moments of the measure of their level sets. As a by product a new generalized Kac-Rice formula (KRF) for the expectation of the measure of these level sets is obtained when the second moment can be uniformly bounded. The conditions involve (i) the differentiability of the trajectories up to a certain order k, (ii) the finiteness of the moments of the k-th partial derivatives of the field up to another order, (iii) the boundedness of the joint density of the field and some of its derivatives. Particular attention is given to the shot noise processes and fields. Other applications include stationary Gaussian processes, Chi-square processes and regularized diffusion processes. AMS2000 Classifications: Primary 60G60. Secondary 60G15.

Percolation of the excursion sets of planar symmetric shot noise fields

We prove the existence of phase transitions in the global connectivity of the excursion sets of planar symmetric shot noise fields. Our main result establishes a phase transition with respect to the level for shot noise fields with symmetric log-concave mark distributions, including Gaussian, uniform, and Laplace marks, and kernels that are positive, symmetric, and have sufficient tail decay. Without the log-concavity assumption we prove a phase transition with respect to the intensity of positive marks.

Lipschitz-Killing curvatures of excursion sets for two-dimensional random fields

In the present paper we study three geometrical characteristics for the excursion sets of a two-dimensional stationary isotropic random field. First, we show that these characteristics can be estimated without bias if the considered field satisfies a kinematic formula, this is for instance the case of fields given by a function of smooth Gaussian fields or of some shot noise fields. By using the proposed estimators of these geometric characteristics, we describe some inference procedures for the estimation of the parameters of the field. An extensive simulation study illustrates the performances of each estimator. Then, we use the Euler characteristic estimator to build a test to determine whether a given field is Gaussian or not, when compared to various alternatives. The test is based on a sparse information, i.e., the excursion sets for two different levels of the field to be tested. Finally, the proposed test is adapted to an applied case, synthesized 2D digital mammograms.

Normal convergence of nonlocalised geometric functionals and shot-noise excursions

This article presents a complete second-order theory for a large class of geometric functionals on homogeneous Poisson input. In particular, the results do not require the existence of a radius of stabilisation. Hence they can be applied to geometric functionals of spatial shot-noise fields excursions such as volume, perimeter, or Euler characteristic (the method still applies to stabilising functionals). More generally, it must be checked that a local contribution to the functional is not strongly affected under a perturbation of the input far away. In this case, the exact asymptotic variance is given, as well as the likely optimal speed of convergence in the central limit theorem. This goes through a general mixing-type condition that adapts nicely to both proving asymptotic normality and that variance is of volume order.

Spectral Efficiency Scaling Laws in Dense Random Wireless Networks With Multiple Receive Antennas

This paper considers large random wireless networks, where transmit-and-receive node pairs communicate within a certain range while sharing a common spectrum. By modeling the spatial locations of nodes as a Poisson point process, analytical expressions for the ergodic spectral efficiency of a typical node pair are derived as a function of the channel state information available at a receiver (CSIR) in terms of relevant system parameters: the density of communication links, the number of receive antennas, the path loss exponent, and the operating signal-to-noise ratio. One key finding is that when the receiver only exploits CSIR for the direct link, the sum spectral efficiency increases linearly with the density, provided the number of receive antennas increases as a certain superlinear function of the density. When each receiver exploits CSIR for a set of dominant interfering links in addition to that of the direct link, the sum spectral efficiency increases linearly with both the density and the path loss exponent if the number of antennas is a linear function of the density. This observation demonstrates that having CSIR for dominant interfering links provides an order gain in the scaling law. It is also shown that this linear scaling holds for direct CSIR when incorporating the effect of the receive antenna correlation, provided that the rank of the spatial correlation matrix scales superlinearly with the density. These scaling laws are derived from integral representations of the distribution of the signal to interference and noise ratio, which are of independent interest and which in turn derived from stochastic geometry and more precisely from the theory of shot noise fields. Simulation results back the scaling laws and the integral representations.

On Scaling Limits of Power Law Shot-Noise Fields

This article studies the scaling limit of a class of shot-noise fields defined on an independently marked stationary Poisson point process and with a power law response function. Under appropriate conditions, it is shown that the shot-noise field can be scaled suitably to have a nondegenerate α-stable limit, as the intensity of the underlying point process goes to infinity. More precisely, finite dimensional distributions are shown to converge, and the finite dimensional distributions of the limiting random field have i.i.d. stable random components. We hence propose to call this limit the α-stable white noise field. Analogous results are also obtained for the extremal shot-noise field that converges to a Fréchet white noise field. Finally, these results are applied to the modeling and analysis of interference fields in large wireless networks.

Central limit theorems for the excursion set volumes of weakly dependent random fields

The multivariate central limit theorems (CLT) for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb{R}^d$ are proved. Special attention is paid to Gaussian and shot noise fields. Formulae for the covariance matrix of the limiting distribution are provided. A statistical version of the CLT is considered as well. Some numerical results are also discussed.

Directionally convex ordering of random measures, shot noise fields, and some applications to wireless communications

Directionally convex ordering is a useful tool for comparing the dependence structure of random vectors, which also takes into account the variability of the marginal distributions. It can be extended to random fields by comparing all finite-dimensional distributions. Viewing locally finite measures as nonnegative fields of measure values indexed by the bounded Borel subsets of the space, in this paper we formulate and study directionally convex ordering of random measures on locally compact spaces. We show that the directionally convex order is preserved under some of the natural operations considered on random measures and point processes, such as deterministic displacement of points, independent superposition, and thinning, as well as independent, identically distributed marking. Further operations on Cox point processes such as position-dependent marking and displacement of points are shown to preserve the order. We also examine the impact of the directionally convex order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions and pair correlation functions, as well as examples, seem to indicate that point processes higher in the directionally convex order cluster more. In our main result we show that nonnegative integral shot noise fields with respect to the directionally convex ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot noise fields appear as key ingredients. We also mention a few pertinent open questions.

Directionally convex ordering of random measures, shot noise fields, and some applications to wireless communications

Directionally convex ordering is a useful tool for comparing the dependence structure of random vectors, which also takes into account the variability of the marginal distributions. It can be extended to random fields by comparing all finite-dimensional distributions. Viewing locally finite measures as nonnegative fields of measure values indexed by the bounded Borel subsets of the space, in this paper we formulate and study directionally convex ordering of random measures on locally compact spaces. We show that the directionally convex order is preserved under some of the natural operations considered on random measures and point processes, such as deterministic displacement of points, independent superposition, and thinning, as well as independent, identically distributed marking. Further operations on Cox point processes such as position-dependent marking and displacement of points are shown to preserve the order. We also examine the impact of the directionally convex order on the second moment properties, in particular on clustering and on Palm distributions. Comparisons of Ripley's functions and pair correlation functions, as well as examples, seem to indicate that point processes higher in the directionally convex order cluster more. In our main result we show that nonnegative integral shot noise fields with respect to the directionally convex ordered random measures inherit this ordering from the measures. Numerous applications of this result are shown, in particular to comparison of various Cox processes and some performance measures of wireless networks, in both of which shot noise fields appear as key ingredients. We also mention a few pertinent open questions.

Extremes of Shot-Noise Fields in the Presence of Regularly Varying Tails with Index $\boldsymbol{\alpha\in(0,1)}$

The paper considers shot-noise fields with regularly varying tail amplitudes with index $\alpha\in (0,1)$. The asymptotic behavior of the supreme of the field over bounded measurable regions growing in the van Hove sense is investigated. The nondegenerate limit law, which was obtained earlier by the author, will be deduced under other assumptions.

General scheme of maxima of sums of independent random variables and its applications

A general scheme of maxima of sums of independent random variables is introduced. We prove a theorem on the convergence of the maxima in probability. We study its applications to large jumps in random walks and to extrema of shot-noise fields in the case of regularly varying tails. Nondegenerate limiting laws are obtained.

Extremes of Shot-Noise Fields in the Presence of Regularly Varying Tails

The paper considers shot-noise fields with regularly varying tailed amplitudes. A~nondegenerate limit law under the linear normalization for maxima of the field over bounded measurable regions growing monotonically in the Van Hove sense is obtained.

Maxima of Subexponential Shot-Noise Fields with Finite Radius of Influence

We study the asymptotic behavior of the maxima of shot-noise fields on bounded measurable domains tending to infinity in the sense of van Hove. It is assumed that the radius of influence is finite and the amplitudes are subexponentially distributed. A nondegenerate limiting distribution for the maxima is obtained.

On the Levy-Baxter theorems for shot-noise fields. III

We establish sufficient conditions for singularity of distributions of shot-noise fields with response functions of a certain form.

On the Levy-Baxter theorems for shot-noise fields. II

We establish sufficient conditions under which shot-noise fields with a response function of a certain form possess the Levy-Baxter property on an increasing parametric set.

On the Levy-Baxter theorems for shot-noise fields. I

On the Levy-Baxter theorems for shot-noise fields. I

Central limit theorem for shot noise fields

One investigates the asymptotic normality of integrals over “unboundedly increasing” sets of a random field, describing a “shot noise” in ℝ d ,d⩾1. One shows how important here are roles played by the growth character of the considered sets and by the dimension d.

Probability Distributions and Level Crossings of Shot Noise Models

This paper deals with shot noise models driven by various point processes. By means of an approximating step response function the distribution of the Poisson shot noise is obtained. The exponential response function case is studied as well, writing down explicit univariate and bivariate distributions. In this case also the crossing problem is analyzed and upcrossing and downcrossing rates are evaluated The last part of the paper is concerned with bivariate shot noise fields in circular regions of the plane, both at inner and boundary points.