Bayesian nonparametric statistical inference for Poisson point processes

Summary A random measure is said to be selected by a weighted gamma prior probability if the values it assigns to disjoint sets are independent gamma random variables with positive multipliers. If the intensity measure of a nonhomogeneous Poisson point process is selected by a weighted gamma prior probability and if a sample is drawn from the Poisson point process having this intensity measure, then the posterior random intensity measure given the observations is also selected by a weighted gamma prior probability. If the measure space is Euclidean and if the true intensity measure is continuous and finite, the centered posterior process, rescaled by the square root of the sample size, will converge weakly in Skorohod topology to a Wiener process subject to a change of time scale.This research was supported in part by the National Science Foundation Grants MCS 77-10376 and MCS 75-14194     

Characterization of Classical and Quantum Poisson Systems by Thinnings and Splittings

There exist several well–known characterizations of Poisson and mixed Poisson point processes (Cox processes) by thinning and splitting procedures. So a point process is necessarily a Cox process if for arbitrary small thinning parameter it can be obtained by a thinning of some other point process [30]. Poisson processes are characterized by the independence of the two random subconfigurations obtained by an independent splitting of the configuration into two parts [11]. For quantum mechanical particle systems beam splittings which are well–known in quantum optics provide analogous procedures. It is shown that coherent states respectively mixtures of them can be characterized in the same way as Poisson processes and Cox processes. Moreover, for the position distributions of these states which are “classical” point processes just the above mentioned characterizations are obtained. As example of mixed coherent states we consider Gaussian states which arise as equilibrium states of ideal Bose gases.     

Mixing of Poisson random measures under interacting transformations

We derive sufficient conditions for the mixing of all orders of interacting transformations of a spatial Poisson point process, under a zero-type condition in probability and a generalized adaptedness condition. This extends a classical result in the case of deterministic transformations of Poisson measures. The approach relies on moment and covariance identities for Poisson stochastic integrals with random integrands.     

Nonsynchronous covariation process and limit theorems

An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales is presented. Two continuous semimartingales are sampled at stopping times in a nonsynchronous manner. Those sampling times possibly depend on the history of the stochastic processes and themselves. The nonsynchronous covariation process converges to the usual quadratic covariation of the semimartingales as the maximum size of the sampling intervals tends to zero. We deal with the case where the limiting variation process of the normalized approximation error is random and prove the convergence to mixed normality, or convergence to a conditional Gaussian martingale. A class of consistent estimators for the asymptotic variation process based on kernels is proposed, which will be useful for statistical applications to high-frequency data analysis in finance. As an illustrative example, a Poisson sampling scheme with random change point is discussed.     

On the exceedance point process for a stationary sequence

Summary It is known that the exceedance points of a high level by a stationary sequence are asymptotically Poisson as the level increases, under appropriate long range and local dependence conditions. When the local dependence conditions are relaxed, clustering of exceedances may occur, based on Poisson positions for the clusters. In this paper a detailed analysis of the exceedance point process is given, and shows that, under wide conditions, any limiting point process for exceedances is necessarily compound Poisson. More generally the possible random measure limits for normalized exceedance point processes are characterized. Sufficient conditions are also given for the existence of a point process limit. The limiting distributions of extreme order statistics are derived as corollaries.This research has been supported by the Air Force Office of Scientific Research Grant No. F 49620 85 C 0144 and the Katholieke Universiteit Leuven     

随机偏好连接图的中心极限定理

我们研究了一类具有随机顶点和边的随机连接图模型, 其中顶点的随机性由一个Poisson 点过程所决定, 边的随机性由一个概率连接函数所决定. 我们得到了带偏好的随机连接图模型的关于所有随机边的长度和的一个中心极限定理.     

Poisson asymptotics for random projections of points on a high-dimensional sphere

Project a collection of points on the high-dimensional sphere onto a random direction. If most of the points are sufficiently far from one another in an appropriate sense, the projection is locally close in distribution to the Poisson point process.     

Multivariate Normal Approximation for Functionals of Random Polytopes

Journal of Theoretical Probability - Consider the random polytope that is given by the convex hull of a Poisson point process on a smooth convex body in $$\mathbb {R}^d$$ . We prove central limit...     

Ergodic properties of crystallization processes

We consider a birth and growth process with germs which are born according to a Poisson point process whose intensity rneasure is invariant under trunslations of the space. The germs can be born in the unoccupied space; then they grow until they occupy the available space. In this general framework, the crystallization process can be characterized by a random field, which assigns to any point of the state space the first time at which this point is reached by a crigstal. Under general conditions on the growth speed and geometric shape of free crystals, we prone that the random field is mixing in the sense of ergodic theory. This result is illustrated by applications to the problem of parameter estimation. Bibliography: 7 titles.     

Identifiability for random translations of Poisson processes

Summary A system in which each point of a stationary Poisson process is subjected to a random displacement, the displacements being independently and identically distributed, is considered. It is shown that the displacement distribution is identifiable if we are given a realization of the original process and the corresponding realization of the displaced process but not the linkage between the two.     

Critical phenomenon for a percolation model

We consider a percolation model which consists of oriented lines placed randomly on the plane. The lines are of random length and at a random angle with respect to the horizontal axis and are placed according to a Poisson point process; the length, angle, and orientation being independent of the underlying Poisson process. We establish a critical behaviour of this model, i.e., percolation occurs for large intensity of the Poisson process and does not occur for smaller intensities. In the special case when the lines are of fixed unit length and are either oriented vertically up or oriented horizontally to the left, with probability p or (1-p), respectively, we obtain a lower bound on the critical intensity of percolation.     

Poisson process with fuzzy rates

In a stochastic homogeneous Poisson process, interarrival times are independent and identically distributed (iid) exponential random variables whose parameter is called the rate of the process. By using fuzzy variables to describe the parameter, a Poisson process whose rates are fuzzy variables is established. Based on the random fuzzy theory, relationship between the renewal number and fuzzy rates is discussed. As an application, a random fuzzy compound Poisson process is investigated.     

Estimation non paramétrique du contour du support d'un processus ponctuel de Poisson

We use one method type kernel estimation with random window for estimate the edge of support of a Poisson point process. The estimator obtained converges uniformly in probability, almost surely, almost completely and converges weakly.     

Strong convergence of multivariate point processes of exceedances

We study the asymptotic behavior of vectors of point processes of exceedances of random thresholds based on a triangular scheme of random vectors. Multivariate maxima w.r.t. marginal ordering may be regarded as a special case. It is proven that strong convergence—that is convergence of distributions w.r.t. the variational distance—of such multivariate point processes holds if, and only if, strong convergence of multivariate maxima is valid. The limiting process of multivariate point processes of exceedances is built by a certain Poisson process. Auxiliary results concerning upper bounds on the variational distance between vectors of point processes are of interest in its own right.The author was supported by the Deutsche Forschungsgemeinschaft.     

Rare Events in Random Geometric Graphs

This work introduces and compares approaches for estimating rare-event probabilities related to the number of edges in the random geometric graph on a Poisson point process. In the one-dimensional setting, we derive closed-form expressions for a variety of conditional probabilities related to the number of edges in the random geometric graph and develop conditional Monte Carlo algorithms for estimating rare-event probabilities on this basis. We prove rigorously a reduction in variance when compared to the crude Monte Carlo estimators and illustrate the magnitude of the improvements in a simulation study. In higher dimensions, we use conditional Monte Carlo to remove the fluctuations in the estimator coming from the randomness in the Poisson number of nodes. Finally, building on conceptual insights from large-deviations theory, we illustrate that importance sampling using a Gibbsian point process can further substantially reduce the estimation variance.

     

Limit behaviors of random connected graphs driven by a Poisson process

We consider a class of random connected graphs with random vertices and random edges with the random distribution of vertices given by a Poisson point process with the intensity n localized at the vertices and the random distribution of the edges given by a connection function. Using the Avram-Bertsimas method constructed in 1992 for the central limit theorem on Euclidean functionals, we find the convergence rate of the central limit theorem process, the moderate deviation, and an upper bound for large deviations depending on the total length of all edges of the random connected graph.     

Bayesian Analysis of a Two-State Markov Modulated Poisson Process

Abstract

We postulate observations from a Poisson process whose rate parameter modulates between two values determined by an unobserved Markov chain. The theory switches from continuous to discrete time by considering the intervals between observations as a sequence of dependent random variables. A result from hidden Markov models allows us to sample from the posterior distribution of the model parameters given the observed event times using a Gibbs sampler with only two steps per iteration.     

Limit laws for coverage in backbone-sensor networks

We study coverage in sensor networks having two types of nodes, namely, sensor nodes and backbone nodes. Each sensor is capable of transmitting information over relatively small distances. The backbone nodes collect information from the sensors. This information is processed and communicated over an ad hoc network formed by the backbone nodes, which are capable of transmitting over much larger distances. We consider two models of deployment for the sensor and backbone nodes. One is a Poisson–Poisson cluster model and the other a dependently thinned Poisson point process. We deduce limit laws for functionals of vacancy in both models using properties of association for random measures.     

Laws of Large Numbers for Observations that Change with Time

Consider a system where units having random magnitude enter according to a nonhomogeneous Poisson process, stay for a random period of time, and then depart. While in the system, a unit's magnitude may change with time. Results are obtained for the strong limiting behavior of the distribution of magnitudes among units present in the system.     

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.