Poisson Process Approximation Research Articles

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).

Multivariate Poisson and Poisson process approximations with applications to Bernoulli sums and -statistics

AbstractThis article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of the Stein equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called $d_\pi$ , stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the point processes on arbitrary collections of disjoint sets. As applications, the multivariate Poisson approximation of the sum of m-dependent Bernoulli random vectors, the Poisson process approximation of point processes of U-statistic structure, and the Poisson process approximation of point processes with Papangelou intensity are considered. Our bounds in $d_\pi$ are as good as those already available in the literature.

Modeling of Aggregated IoT Traffic and Its Application to an IoT Cloud

As the Internet of Things (IoT) continues to gain traction in telecommunication networks, a very large number of devices are expected to be connected and used in the near future. In order to appropriately plan and dimension the network, as well as the back-end cloud systems and the resulting signaling load, traffic models are employed. These models are designed to accurately capture and predict the properties of IoT traffic in a concise manner. To achieve this, Poisson process approximations, based on the Palm–Khintchine theorem, have often been used in the past. Due to the scale (and the difference in scales in various IoT networks) of the modeled systems, the fidelity of this approximation is crucial, as, in practice, it is very challenging to accurately measure or simulate large-scale IoT deployments. The main goal of this paper is to understand the level of accuracy of the Poisson approximation model. To this end, we first survey both common IoT network properties and network scales as well as traffic types. Second, we explain and discuss the Palm–Khintiche theorem, how it is applied to the problem, and which inaccuracies can occur when using it. Based on this, we derive guidelines as to when a Poisson process can be assumed for aggregated periodic IoT traffic. Finally, we evaluate our approach in the context of an IoT cloud scaler use case.

Refined total variation bounds in the multivariate and compound Poisson approximation

We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better structure than those from the literature. A numerical example illustrates the usefulness of the bounds, and an application in the Poisson process approximation is given. The proofs use arguments from Kerstan (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964) 173-179) and Roos (J. Multivariate Anal. 69 (1999) 120-134) in combination with new smoothness inequalities, which could be of independent interest.

Functional Poisson approximation in Kantorovich–Rubinstein distance with applications to U-statistics and stochastic geometry

A Poisson or a binomial process on an abstract state space and a symmetric function $f$ acting on $k$-tuples of its points are considered. They induce a point process on the target space of $f$. The main result is a functional limit theorem which provides an upper bound for an optimal transportation distance between the image process and a Poisson process on the target space. The technical background are a version of Stein's method for Poisson process approximation, a Glauber dynamics representation for the Poisson process and the Malliavin formalism. As applications of the main result, error bounds for approximations of U-statistics by Poisson, compound Poisson and stable random variables are derived, and examples from stochastic geometry are investigated.

Approximating dependent rare events

In this paper we give a historical account of the development of Poisson approximation using Stein's method and present some of the main results. We give two recent applications, one on maximal arithmetic progressions and the other on bootstrap percolation. We also discuss generalisations to compound Poisson approximation, Poisson process approximation and multivariate Poisson approximation, and state a few open problems.

On the asymptotics of locally dependent point processes

We investigate a family of approximating processes that can capture the asymptotic behaviour of locally dependent point processes. We prove two theorems presented to accommodate respectively the positively and negatively related dependent structures. Three examples are given to illustrate that our approximating processes can circumvent the technical difficulties encountered in compound Poisson process approximation (see Barbour and Månsson (2002) [10]) and our approximation error bound decreases when the mean number of the random events increases, in contrast to the increasing of bounds for compound Poisson process approximation.

Removing logarithms in Poisson approximation to the partial sum process of a Markov chain

He and Xia (1997, Stochastic Processes Appl. 68, pp. 101–111) gave some error bounds for a Wasserstein distance between the distributions of the partial sum process of a Markov chain and a Poisson point process on the positive half-line. However, all these bounds increase logarithmically with the mean of the Poisson point process. In this paper, using the coupling method and a general deep result for estimating the errors of Poisson process approximation in Brown and Xia (2001, Ann. Probab. 29, pp. 1373–1403), we give a new error bound for the above Wasserstein distance. In contrast to the previous results of He and Xia (1997), our new error bound has no logarithm anymore and it is bounded and asymptotically remains constant as the mean increases.

Joint Vertex Degrees in the Inhomogeneous Random Graph Model ℊ(n,{pij })

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

Joint Vertex Degrees in the Inhomogeneous Random Graph Model ℊ(n,{pij})

In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly inhomogeneous distribution, only when the degrees are large can we reasonably approximate the joint counts as independent. The proofs are based on Stein's method and the Stein-Chen method with a new size-biased coupling for such inhomogeneous random graphs, and, hence, bounds on the distributional distance are obtained. Finally, we illustrate that apparent (pseudo-)power-law-type behaviour can arise in such inhomogeneous networks despite not actually following a power-law degree distribution.

Poisson process approximation for dependent superposition of point processes

Although the study of weak convergence of superpositions of point processes to the Poisson process dates back to the work of Grigelionis in 1963, it was only recently that Schuhmacher [Stochastic Process. Appl. 115 (2005) 1819–1837] obtained error bounds for the weak convergence. Schuhmacher considered dependent superposition, truncated the individual point processes to 0–1 point processes and then applied Stein’s method to the latter. In this paper, we adopt a different approach to the problem by using Palm theory and Stein’s method, thereby expressing the error bounds in terms of the mean measures of the individual point processes, which is not possible with Schuhmacher’s approach. We consider locally dependent superposition as a generalization of the locally dependent point process introduced in Chen and Xia [Ann. Probab. 32 (2004) 2545–2569] and apply the main theorem to the superposition of thinned point processes and of renewal processes.

Stein’s method and Poisson process approximation for a class of Wasserstein metrics

Based on Stein's method, we derive upper bounds for Poisson process approximation in the $L_1$-Wasserstein metric $d_2^{(p)}$, which is based on a slightly adapted $L_p$-Wasserstein metric between point measures. For the case $p=1$, this construction yields the metric $d_2$ introduced in [Barbour and Brown Stochastic Process. Appl. 43 (1992) 9--31], for which Poisson process approximation is well studied in the literature. We demonstrate the usefulness of the extension to general $p$ by showing that $d_2^{(p)}$-bounds control differences between expectations of certain $p$th order average statistics of point processes. To illustrate the bounds obtained for Poisson process approximation, we consider the structure of 2-runs and the hard core model as concrete examples.

A new metric between distributions of point processes

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric d̅ 1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about d̅ 1 and its induced Wasserstein metric d̅ 2 for point process distributions are given, including examples of useful d̅ 1-Lipschitz continuous functions, d̅ 2 upper bounds for the Poisson process approximation, and d̅ 2 upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of d̅ 1 in applications.

A new metric between distributions of point processes

Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metricd̅1that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results aboutd̅1and its induced Wasserstein metricd̅2for point process distributions are given, including examples of usefuld̅1-Lipschitz continuous functions,d̅2upper bounds for the Poisson process approximation, andd̅2upper and lower bounds between distributions of point processes of independent and identically distributed points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential ofd̅1in applications.

On the Role of Inherent Structures in Glass-Forming Materials: I. The Vitrification Process

In this work, we investigate the role of inherent structures in the vitrification process of glass-forming materials by using a two-component Lennard-Jones mixture. We start with a simplified model that describes the dynamics of the atomistic system as a Poisson process consisting of a series of transitions from one potential energy minimum (inherent structure) to another, the rate of individual transitions being described by a first-order kinetic law. We investigate the validity of this model by comparing the mean square displacement resulting from atomistic molecular dynamics (MD) trajectories with the corresponding mean square displacement based on inherent structures. Furthermore, in the case of vitrification via stepwise cooling, we identify the role of the potential energy landscape in determining the properties of the resulting glass. Interestingly, the cooling rate is not sufficient to define the resulting glass in a stepwise cooling process, because the time spent by the system at different temperatures (length of the steps) has a highly nonlinear impact on the properties of the resulting glass. In contrast to previous investigations of supercooled liquids, we focus on a range of temperatures close to and below the glass transition temperature, where the use of MD is incapable of producing equilibrated samples of the metastable supercooled state. Our aim is to develop a methodology that enables mapping the dynamics under these conditions to a coarse-grained first-order kinetic model based on the Poisson process approximation. This model can be used in order to extend our dynamical sampling ability to much broader time scales and therefore allow us to create computer glasses with cooling rates closer to those used experimentally. In a continuation to this work, we provide the mathematical formulation for lifting the coarse-grained Poisson process model and reproducing the full dynamics of the atomistic system.

A polynomial birth–death point process approximation to the Bernoulli process

We propose a class of polynomial birth–death point processes (abbreviated to PBDP) Z ≔ ∑ i = 1 Z δ U i , where Z is a polynomial birth–death random variable defined in [T.C. Brown, A. Xia, Stein’s method and birth–death processes, Ann. Probab. 29 (2001) 1373–1403], U i ’s are independent and identically distributed random elements on a compact metric space, and U i ’s are independent of Z . We show that, with two appropriately chosen parameters, the error of PBDP approximation to a Bernoulli process is of the order O ( n − 1 / 2 ) with n being the number of trials in the Bernoulli process. Our result improves the performance of Poisson process approximation, where the accuracy is mainly determined by the rarity (i.e. the success probability) of the Bernoulli trials and the dependence on sample size n is often not explicit in the bound.

Compound Poisson process approximation for locally dependent real-valued random variables via a new coupling inequality

We present a general and quite simple upper bound for the total variation distance d TV between any stochastic process ( X i ) i ∈Γ defined over a countable space Γ , and a compound Poisson process on Γ . This result is sufficient for proving weak convergence for any functional of the process ( X i ) i ∈Γ when the real-valued X i are rarely non-zero and locally dependent. Our result is established after introducing and employing a generalization of the basic coupling inequality. Finally, two simple examples of application are presented in order to illustrate the applicability of our results.

Distance estimates for dependent superpositions of point processes

In this article, superpositions of possibly dependent point processes on a general space X are considered. Using Stein's method for Poisson process approximation, an estimate is given for the Wasserstein distance d 2 between the distribution of such a superposition and an appropriate Poisson process distribution. This estimate is compared to a modern version of Grigelionis’ theorem, and to results of Banys [Lecture Notes in Statistics, vol. 2, Springer, New York, 1980, pp. 26–37], Arratia et al. [Ann. Probab. 17 (1989) 9–25] and Barbour et al. [Poisson Approximation, Oxford University Press, Oxford, 1992]. Furthermore, an application to a spatial birth–death model is presented.

IMPROVED RESULTS ON POISSON PROCESS APPROXIMATION IN JACKSON NETWORKS

Melamed (1979) proved that for an open migration process, a necessary and sufficient condition for the equilibrium flow along a link to be Poissonian is the absence of loops: no customer can travel along the link more than once. Barbour and Brown (1996) quantified the statement by allowing the customers a small probability of travelling along the link more than once and proved Poisson process approximation theorems analogous to Melamed's Theorem. Amongst the three bounds presented in Barbour and Brown (1996), the one in terms of the Wasserstein metric is of particular interest since it reveals more insightful information about the closeness between the process of flows and an approximating Poisson process, and it is small when the parameter of the system is small, except a logarithmic factor in terms of time in which the flows are considered. The bound was later improved by Brown, Weinberg and Xia (2000) who showed that the logarithmic factor in terms of time can be lifted at the cost of an extra parameter being introduced into the bound. In this paper, we present a new bound which simplifies and sharpens the bounds in the above-mentioned two papers and compare the performance of these bounds for a simple open migration process.

Stein’s method, Palm theory and Poisson process approximation

The framework of Stein's method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem \refimportantproposition) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Wasserstein pseudometric is introduced for measuring the accuracy of point process approximation. The pseudometric provides a generalization of many metrics used so far, including the total variation distance for random variables and the Wasserstein metric for processes as in Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9-31]. Also, through the pseudometric, approximation for certain point processes on a given carrier space is carried out by lifting it to one on a larger space, extending an idea of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990) 403-434]. The error bound in the general result is similar in form to that for Poisson approximation. As it yields the Stein factor 1/\lambda as in Poisson approximation, it provides good approximation, particularly in cases where \lambda is large. The general result is applied to a number of problems including Poisson process modeling of rare words in a DNA sequence.