Binomial Point Processes Research Articles

Coverage Analysis of SAGIN With Sectorized Beam Pattern Under Shadowed-Rician Fading Channels

Space-air-ground integrated networks (SAGIN) have become a research hotspot facing the next generation of communications. The theoretical analysis for non-terrestrial networks (NTN) is significant before applying them in practical scenarios, but the existing works failed to provide a general analysis approach for NTN. Against this background, multiple satellites and civil aircrafts (CAs) are modeled as 3-D binomial point processes (BPPs) in the given finite space in this paper, and we desire to investigate the coverage performance of downlink CA augmented-SAGIN (CAA-SAGIN). Considering the sectorized beam pattern of platforms, we provide a detailed analysis of the different distributions of the serving and interfering platforms and derive the Laplace transform of the interference under shadowed-Rician fading channels. Then, the exact and closed-form expressions are obtained for the general cases with interference and the particular cases without interference via stochastic geometry. The approximations and boundary values are derived by adopting the existing mathematical theories. We analyze the effects of different parameters on the coverage probability of satellite and CA networks, and prove the validity of the derived analytical expressions, approximations, and bounds. Moreover, this work paves the way from the system level to exploit the generic coverage performance of NTN.

Cell-Free Massive MIMO With Finite Fronthaul Capacity: A Stochastic Geometry Perspective

In this work, we analyze the downlink performance of a cell-free massive multiple-input-multiple-output system with finite capacity fronthaul links between the centralized baseband unit and the access point (APs). Conditioned on the user and AP locations, we first derive an achievable rate for a randomly selected user in the network that captures the effect of finite fronthaul capacity as a compression error. From this expression, we establish that for the traditional cell-free architecture where each AP serves all the users in the network, the achievable rate becomes zero as the network size grows. Hence, to have a meaningful analysis, for the traditional architecture, we model the user and AP locations as two independent binomial point processes over a finite region and provide an accurate theoretical result to determine the user rate coverage. In contrast, for an asymptotically large network, we consider a user-centric architecture where each user in the network is served by a specified number of nearest APs that limits the fronthaul load. For this architecture, we model the AP and user locations as two independent Poisson point processes (PPPs). Since the rate expression is a function of the number of users served by an AP, we statistically characterize the load in terms of the number of users per AP. As the exact derivation of the probability mass function of the load is intractable, we first present the exact expressions for the first two moments of the load. Next, we approximate the load as a negative binomial random variable through the moment matching method. Using the load results along with appropriate distance distributions of a PPP, we present an accurate theoretical expression for the rate coverage of the typical user. From the analyses, we conclude that for the traditional architecture when the AP transmit power is relatively high, a more collocated antenna deployment is preferred. Further, for the user-centric architecture, the energy efficiency of the system is a concave function of the number of antennas per AP.

Large deviations for the volume of k-nearest neighbor balls

This paper develops the large deviations theory for the point process associated with the Euclidean volume of k-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of k-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of M0-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most k in a random geometric graph in the dense regime.

Poisson process approximation under stabilization and Palm coupling

We present new Poisson process approximation results for stabilizing functionals of Poisson and binomial point processes. These functionals are allowed to have an unbounded range of interaction and encompass many examples in stochastic geometry. Our bounds are derived for the Kantorovich-Rubinstein distance using the generator approach to Stein's method. We give different types of bounds for different point processes. While some of our bounds are given in terms of coupling of the point process with its Palm version, the others are in terms of the local dependence structure formalized via the notion of stabilization. We provide two supporting examples for our new framework - one is for Morse critical points of the distance function, and the other is for large k-nearest neighbor balls. Our bounds considerably extend the results in Barbour and Brown (1992), Decreusefond, Schulte and Thale (2016) and Otto (2020).

Modeling and Analysis of mmWave UAV Swarm Networks: A Stochastic Geometry Approach

In order to evaluate the impacts of swarm distribution characteristics of unmanned aerial vehicles (UAVs) and the complexity of millimeter wave (mmWave) propagations on the coverage performance of UAV networks, this paper develops a stochastic geometry-based approach for the modeling and analysis of single- and multi-swarm UAV networks with mmWave communications. We first utilize binomial point processes (BPPs) to model the locations of UAVs in a finite area as a single-swarm UAV network and allow for arbitrary user locations. Then, we define multi-BPPs and extend our model to multi-swarm UAV networks. We consider an open-access strategy, where users access the closest UAV in all swarms, and derive the distance distributions of serving and interfering UAVs. Moreover, a three-dimensional (3D) blockage model and a 3D sectorized antenna model are introduced to completely depict the characteristics of air-to-ground channel in mmWave band. Based on the distance distributions and the Laplace transform of interference, coverage probabilities in considered networks are derived. Our results reveal that, there exists optimal height and density of UAV swarms that maximize the coverage probability. When densely deployed, the interference from inter-swarm cannot be negligible in the multi-swarm UAV network.

When Satellites Work as Eavesdroppers

This paper considers <i>satellite eavesdroppers</i> in uplink satellite communication systems where the eavesdroppers are randomly distributed at arbitrary altitudes according to homogeneous binomial point processes and attempt to overhear signals that a ground terminal transmits to a serving satellite. Non-colluding eavesdropping satellites are assumed, i.e., they do not cooperate with each other, so that their received signals are not combined but are decoded individually. Directional beamforming with two types of antennas: fixed- and steerable-beam antennas, is adopted at the eavesdropping satellites. The possible distribution cases for the eavesdropping satellites and the distributions of the distances between the terminal and the satellites are analyzed. The distributions of the signal-to-noise ratios (SNRs) at both the serving satellite and the most detrimental eavesdropping satellite are derived as closed-form expressions. The ergodic and outage secrecy capacities of the systems are derived with the secrecy outage probability using the SNR distributions. Simpler approximate expressions for the secrecy performance are obtained based on the Poisson limit theorem, and asymptotic analyses are also carried out in the high-SNR regime. Monte-Carlo simulations verify the analytical results for the secrecy performance. The analytical results are expected to be used to evaluate the secrecy performance and design secure satellite constellations by considering the impact of potential threats from malicious satellite eavesdroppers.

Transport inequalities for random point measures

We derive transport-entropy inequalities for mixed binomial point processes, and for Poisson point processes. We show that when the finite intensity measure satisfies a Talagrand transport inequality, the law of the point process also satisfies a Talagrand type transport inequality. We also show that a Poisson point process (with arbitrary σ-finite intensity measure) always satisfies a universal transport-entropy inequality à la Marton. We explore the consequences of these inequalities in terms of concentration of measure and modified logarithmic Sobolev inequalities. In particular, our results allow one to extend a deviation inequality by Reitzner [33], originally proved for Poisson random measures with finite mass.

Meta Distribution of Downlink SIR for Binomial Point Processes

The meta distribution (MD) of the signal to interference ratio (SIR) extends stochastic geometry analysis from spatial averages to reveals find-grained information about the network performance. There have been several efforts to establish the MD framework for the Poisson point process (PPP) and other ergodic point processes. However, the MD analysis for finite point processes is overlooked. In this letter, we develop the MD of the binomial point process (BPP), which is practical for cases with a priori knowledge about the number of devices as well as their geographical spatial existence. For such finite models, we define the MD as a location-dependent likelihood of a receiver to achieve a required SIR with a probability more than a predefined threshold. This letter also extends the MD of the BPP to find the MD of finite PPP and verifies the convergence of the newly derived MD to the ergodic PPP's MD. The obtained analytical derivations are validated using Monte-Carlo simulations.

Joint Distribution of the Number of Vertices and the Area of Convex Hulls Generated by a Uniform Distribution in a Convex Polygon

A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process

Nearest Neighbor and Contact Distance Distribution for Binomial Point Process on Spherical Surfaces

This letter characterizes the statistics of the contact distance and the nearest neighbor (NN) distance for binomial point processes (BPP) spatially-distributed on spherical surfaces. We consider a setup of $n$ concentric spheres, with each sphere $S_{k}$ has a radius $r_{k}$ and $N_{k}$ points that are uniformly distributed on its surface. For that setup, we obtain the cumulative distribution function (CDF) of the distance to the nearest point from two types of observation points: (i) the observation point is not a part of the point process and located on a concentric sphere with a radius $r_{e} , which corresponds to the contact distance distribution, and (ii) the observation point belongs to the point process, which corresponds to the nearest-neighbor (NN) distance distribution.

Downlink Coverage and Rate Analysis of Low Earth Orbit Satellite Constellations Using Stochastic Geometry

As low Earth orbit (LEO) satellite communication systems are gaining increasing popularity, new theoretical methodologies are required to investigate such networks' performance at large. This is because deterministic and location-based models that have previously been applied to analyze satellite systems are typically restricted to support simulations only. In this paper, we derive analytical expressions for the downlink coverage probability and average data rate of generic LEO networks, regardless of the actual satellites' locality and their service area geometry. Our solution stems from stochastic geometry, which abstracts the generic networks into uniform binomial point processes. Applying the proposed model, we then study the performance of the networks as a function of key constellation design parameters. Finally, to fit the theoretical modeling more precisely to real deterministic constellations, we introduce the effective number of satellites as a parameter to compensate for the practical uneven distribution of satellites on different latitudes. In addition to deriving exact network performance metrics, the study reveals several guidelines for selecting the design parameters for future massive LEO constellations, e.g., the number of frequency channels and altitude.

On central limit theorems in stochastic geometry for add-one cost stabilizing functionals

We establish central limit theorems for general functionals on binomial point processes and their Poissonized version, which extends the results of Penrose–Yukich (Ann. Appl. Probab. 11(4), 1005–1041 (2001)) to the inhomogeneous case. Here functionals are required to be strongly stabilizing for add-one cost on homogeneous Poisson point processes and to satisfy some moments conditions. As an application, a central limit theorem for Betti numbers of random geometric complexes in the subcritical regime is derived.

Strong Law of Large Numbers for Betti Numbers in the Thermodynamic Regime

We establish the strong law of large numbers for Betti numbers of random \v{C}ech complexes built on $\mathbb R^N$-valued binomial point processes and related Poisson point processes in the thermodynamic regime. Here we consider both the case where the underlying distribution of the point processes is absolutely continuous with respect to the Lebesgue measure on $\mathbb R^N$ and the case where it is supported on a $C^1$ compact manifold of dimension strictly less than $N$. The strong law is proved under very mild assumption which only requires that the common probability density function belongs to $L^p$ spaces, for all $1\leq p < \infty$.

New Berry–Esseen bounds for functionals of binomial point processes

We obtain explicit Berry–Esseen bounds in the Kolmogorov distance for the normal approximation of nonlinear functionals of vectors of independent random variables. Our results are based on the use of Stein’s method and of random difference operators, and generalise the bounds obtained by Chatterjee (2008), concerning normal approximations in the Wasserstein distance. In order to obtain lower bounds for variances, we also revisit the classical Hoeffding decompositions, for which we provide a new proof and a new representation. Several applications are discussed in detail: in particular, new Berry–Esseen bounds are obtained for set approximations with random tessellations, as well as for functionals of coverage processes.

Normal approximation for stabilizing functionals

We establish presumably optimal rates of normal convergence with respect to the Kolmogorov distance for a large class of geometric functionals of marked Poisson and binomial point processes on general metric spaces. The rates are valid whenever the geometric functional is expressible as a sum of exponentially stabilizing score functions satisfying a moment condition. By incorporating stabilization methods into the Malliavin-Stein theory, we obtain rates of normal approximation for sums of stabilizing score functions which either improve upon existing rates or are the first of their kind. Our general rates hold for functionals of marked input on spaces more general than full-dimensional subsets of $\mathbb{R}^d$, including $m$-dimensional Riemannian manifolds, $m\leq d$. We use the general results to deduce improved and new rates of normal convergence for several functionals in stochastic geometry, including those whose variances re-scale as the volume or the surface area of an underlying set. In particular, we improve upon rates of normal convergence for the $k$-face and $i$th intrinsic volume functionals of the convex hull of Poisson and binomial random samples in a smooth convex body in dimension $d\geq 2$. We also provide improved rates of normal convergence for statistics of nearest neighbors graphs and high-dimensional data sets, the number of maximal points in a random sample, estimators of surface area and volume arising in set approximation via Voronoi tessellations, and clique counts in generalized random geometric graphs.

A remark on the convergence of Betti numbers in the thermodynamic regime

The convergence of the expectations of Betti numbers of Čech complexes built on binomial point processes in the thermodynamic regime is established.

Percolation and limit theory for the poisson lilypond model

AbstractThe lilypond model on a point process in d ‐space is a growth‐maximal system of non‐overlapping balls centred at the points. We establish central limit theorems for the total volume and the number of components of the lilypond model on a sequence of Poisson or binomial point processes on expanding windows. For the lilypond model over a homogeneous Poisson process, we give subexponentially decaying tail bounds for the size of the cluster at the origin. Finally, we consider the enhanced Poisson lilypond model where all the balls are enlarged by a fixed amount (the enhancement parameter), and show that for d &gt; 1 the critical value of this parameter, above which the enhanced model percolates, is strictly positive. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

Normal approximation for coverage models over binomial point processes

We give error bounds which demonstrate optimal rates of convergence in the CLT for the total covered volume and the number of isolated shapes, for germ-grain models with fixed grain radius over a binomial point process of n points in a toroidal spatial region of volume n. The proof is based on Stein’s method via size-biased couplings.

Strictly stable distributions on convex cones

Using the LePage representation, a symmetric alpha-stable random element in Banach space B with alpha from (0,2) can be represented as a sum of points of a Poisson process in B. This point process is union-stable, i.e. the union of its two independent copies coincides in distribution with the rescaled original point process. This shows that the classical definition of stable random elements is closely related to the union-stability property of point processes. These concepts make sense in any convex cone, i.e. in a semigroup equipped with multiplication by numbers, and lead to a construction of stable laws in general cones by means of the LePage series. We prove that random samples (or binomial point processes) in rather general cones converge in distribution in the vague topology to the union-stable Poisson point process. This convergence holds also in a stronger topology, which implies that the sums of points converge in distribution to the sum of points of the union-stable point process. Since the latter corresponds to a stable law, this yields a limit theorem for normalised sums of random elements with alpha-stable limit for alpha from (0,1). By using the technique of harmonic analysis on semigroups we characterise distributions of alpha-stable random elements and show how possible values of the characteristic exponent alpha relate to the properties of the semigroup and the corresponding scaling operation, in particular, their distributivity properties. It is shown that several conditions imply that a stable random element admits the LePage representation. The approach developed in the paper not only makes it possible to handle stable distributions in rather general cones (like spaces of sets or measures), but also provides an alternative way to prove classical limit theorems and deduce the LePage representation for strictly stable random vectors in Banach spaces.

Multivariate spatial central limit theorems with applications to percolation and spatial graphs

Suppose X = (X x , x in Z d ) is a family of i.i.d. variables in some measurable space, B 0 is a bounded set in R d , and for t > 1, H t is a measure on t B 0 determined by the restriction of X to lattice sites in or adjacent to t B 0 . We prove convergence to a white noise process for the random measure on B 0 given by t -d/2 (H t (tA) - EH t (tA)) for subsets A of B 0 , as t becomes large. subject to H satisfying a stabilization condition (whereby the effect of changing X at a single site x is local) but with no assumptions on the rate of decay of correlations. We also give a multivariate central limit theorem for the joint distributions of two or more such measures H t , and adapt the result to measures based on Poisson and binomial point processes. Applications given include a white noise limit for the measure which counts clusters of critical percolation, a functional central limit theorem for the empirical process of the edge lengths of the minimal spanning tree on random points, and central limit theorems for the on-line nearest-neighbor graph.