泊松点过程的若干大偏差估计

本文给出泊松点过程下列三种极限行为的大偏差估计：（１）高密度情形；（２）低密度情形和（３）标度变换下极限情形。     

A New Approach to Poisson Approximation of Simple Point Processes Using Compensators

Consider a simple point process N on the line, and let be its compensator. We use a result of Kallenberg (1990, Probab. Theory Relat. Fields 86, 167–202) to give a new approach to estimate the total variation distance between the distributions of N and that of a Poisson process when has small jump sizes.     

Second-Order Efficient Test for Inhomogeneous Poisson Processes

We observe a realization X ⁽ⁿ⁾ of a Poisson process on the set with intensity function depending on the unknown real parameter . Based on X ⁽ⁿ⁾ we test simple null hypothesis against one sided alternative for given . We improve the level of the well-known locally asymptotically uniformly most powerful (LAUMP) test by using the Edgeworth type expansion for stochastic integral. We show that the improved test is second-order efficient under certain regularity conditions.      

On Rescaled Poisson Processes and the Brownian Bridge

The process obtained by rescaling a homogeneous Poisson process by the maximum likelihood estimate of its intensity is shown to have surprisingly strong self-correcting behavior. Formulas for the conditional intensity and moments of the rescaled Poisson process are derived, and its behavior is demonstrated using simulations. Relationships to the Brownian bridge are explored, and implications for point process residual analysis are discussed.     

Approximation of Hunt Processes by Multivariate Poisson Processes

We prove that arbitrary Hunt processes on a general state space can be approximated by multivariate Poisson processes starting from each point of the state space. The key point is that no additional regularity assumption on the state space and on the underlying transition semigroup is used.     

On Estimation of Poisson Intensity Functions

Under the presence of only one realization, we consider a computationally simple algorithm for estimating the intensity function of a Poisson process with exponential quadratic and cyclic of fixed frequency trends. We argue that the algorithm can successfully be used to estimate any Poisson intensity function provided that it has a parametric form.     

离散抽样OU-复合Poisson过程的参数矩估计

本文研究基于离散观测的正复合Poisson过程驱动OU型过程的参数估计. 通过矩估计给出了过程平稳分布参数的估计量, 并得到了估计量的相合性和渐近正态性. 进一步, 将矩估计的方法和结论推广到叠加过程的情况.     

On Random Sets Connected to the Partial Records of Poisson Point Process

Random intervals are constructed from partial records in a Poisson point process in ]0,[×]0,[. These are used to cover partially [0,[; the purpose of this work is to study the random set that is left uncovered. We show that enjoys the regenerative property and identify its distribution in terms of the characteristics of the Poisson point process. As an application we show that is almost surely a fractal set and we calculate its dimension.     

Continuity and Boundedness of Infinitely Divisible Processes: A Poisson Point Process Approach

Sufficient conditions for boundedness and continuity are obtained for stochastically continuous infinitely divisible processes, without Gaussian component, {Y(t),t T}, where T is a compact metric space or pseudo-metric space. Such processes have a version given by Y(t)=X(t)+b(t),tT where b is a deterministic drift function and

膨胀的Poisson过程及其在金融中的应用

本文定义了一种增量不独立的纯跳过程,称为膨胀的Poisson过程.采用了一个n跳过程来描述股票市场价格运动的规律,并构造了一个货币市场投资组合使得它的市场价值在指定时刻与股票价格相等,且该投资组合的收益被分解成为一个确定性的项和一个膨胀的Poisson项之和.证明了投资投票市场风险大于投资货币市场风险.     

A class of renewal Interrupted Poisson Processes and applications to queueing systems

Switched Poisson Processes and Interrupted Poisson Processes are often employed to characterize traffic streams in distributed computer and communications systems, especially in investigations of overflow processes in telecommunication networks. With these processes, input streams having inter-segment correlations and high variance as well as state-dependent traffic can properly be modelled. In this paper we first derive an approximation method to describe the Generalized Switched Poisson processes in conjunction with a renewal assumption. As a special case of this class of processes, the class of Interrupted Poisson processes is also included in the investigation. As a result, a generalization of the well-known class of Interrupted Poisson processes is obtained. It is shown that the renewal property is also given for this general class of Interrupted Poisson processes having generally distributed off-phase. To illustrate the accuracy of the presented renewal approximation of Generalized Switched Poisson processes and to show the major properties of the General Interrupted Poisson processes, applications to some basic queueing systems are discussed by means of numerical results.This work was done while the author was with Institute of Communications Switching and Data Technics, University of Stuttgart, Seidenstrasse 36, D-7000 Stuttgart 1, FRG.     

Poisson Traffic Processes in RGSMP and Generalized Networks

1.Introductiontrafficprocessesinqueueingnetworksareanimportantoperatingfacetofsuchmodels,aswellasvaluableinthestudyofvaliddecompositionsofnetworks.IfwefindsometrafficprocessesinanetworkPoisson,thenitoftenrendersthemathematicalanalysistractable.Generalized…     

Classification of Mixtures of Spatial Point Processes via Partial Bayes Factors

Motivated by the problem of minefield detection, we investigate the problem of classifying mixtures of spatial point processes. In particular we are interested in testing the hypothesis that a given dataset was generated by a Poisson process versus a mixture of a Poisson process and a hard-core Strauss process. We propose testing this hypothesis by comparing the evidence for each model by using partial Bayes factors. We use the term partial Bayes factor to describe a Bayes factor, a ratio of integrated likelihoods, based on only part of the available information, namely that information contained in a small number of functionals of the data. We applied our method to both real and simulated data, and considering the difficulty of classifying these point patterns by eye, our approach overall produced good results.     

Optimal Upper and Lower Bounds for the Upper Tails of Compound Poisson Processes

A compound Poisson process is of the form where Z, Z ₁, Z ₂, are arbitrary i.i.d. random variables and N is an independent Poisson random variable with parameter . This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate . The truncation level introduced depends only on and Z and not on the overall exceedance level a.     

Point processes characterized by their one dimensional distributions

It is well-known that the distribution of a point process defined on a carrier space is uniquely characterised by its finite dimensional joint distributions of counts on disjoint subsets of . In this note, we investigate the common structure of point processes whose distributions are specified by their one dimensional distributions. We also show that, if is such a point process, then a sequence of point processes { _n } converges in distribution to if and only if { _n (B)} converges in distribution to (B) for a suitably rich class of sets B. Supported by ARC Discovery project number DP0209179 Mathmatics Subject Classification (2000):Primary 60G55; Secondary 60E05, 60B10 AcknowledgementI would like to thank a referee for his valuable suggestions on the presentation of this paper.     

防洪风险分析中的一个二元标值犘狅犻狊狊狅狀点过程模型

该文将洪水的大小和持续时间作为防洪设施的工程风险中不可忽略的因素,提出了以洪水的大小和持续时间为标值的二元标值Ｐｏｉｓｓｏｎ点过程模型,给出了防洪综合风险率的计算公式,并进行了实例计算．     

Approximation and Simulation of the Distributions of Scan Statistics for Poisson Processes in Higher Dimensions

Given a Poisson process in two or three dimensions, we are interested in the scan statistic, i.e. the largest number of points contained in a translate of a fixed scanning set restricted to lie inside a rectangular area. The distribution of the scan statistic is accurately approximated for rectangular scanning sets, using a technique that is also extended to higher dimensions. The accuracy of the approximation is checked through simulation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Estimation for nonhomogeneous Poisson processes from aggregated data

A well-known heuristic for estimating the rate function or cumulative rate function of a nonhomogeneous Poisson process assumes that the rate function is piecewise constant on a set of data-independent intervals. We investigate the asymptotic (as the amount of data grows) behavior of this estimator in the case of equal interval widths, and show that it can be transformed into a consistent estimator if the interval lengths shrink at an appropriate rate as the amount of data grows.     

A Generalized Renewal Equation for Perturbed Compound Poisson Processes with Two-Sided Jumps

Abstract

In this article, we study the discounted penalty at ruin in a perturbed compound Poisson model with two-sided jumps. We show that it satisfies a renewal equation under suitable conditions and consider an application of this renewal equation to study some perpetual American options. In particular, our renewal equation gives a generalization of the renewal equation in Gerber and Landry [2 Gerber , H.U. , and Landry , B. 1998 . On the discounted penalty at ruin in a jump-diffusion and the perpetual put option . Insurance: Mathematics and Economics 22 : 263 – 276 .[Crossref], [Web of Science ®] , [Google Scholar]] where only downward jumps are allowed.     

Poisson limits for a hard-core clustering model

Let X_1n,…,X_>nn denote the locations of n points in a bounded, γ-dimensional, Euclidean region D_n which has positive γ-dimensional Lebesgue measure μ(D_n). Let {Y_n(r): r > 0} be the interpoint distance process for these points where Y_n(r) is the number of pairs of points(X_in, X_in) which with i < j have Euclidean distance 6X_in ? X_>in6 < r. In this article we study the limiting distribution of Y_n(r) when n → ∞ and μ(D_n) → ∞, and the joint density of X_1n,…,X_nnis of the form

where r₀ is a positive constant and C_n is a normalizing constant. These joint densities modify the Strauss [11] clustering model densities by introducing a hard-core component (no two points can have 6X_in ? X_in6 < r₀) found in the Matérn [4] models. In our main result we show that the interpoint distance process converges to a non-homogeneous Poisson process for r values in a bounded interval 0 < r₀ < r < r₀₀ provided sparseness conditions discussed by Saunders and Funk [9] hold. The sparseness conditions which require $\text{μ(D}{}_{\mathrm{n}}\text{)}\text{n}{}^2$ converges to a positive constant and the boundary of D_n is negligible are essentially equivalent to requiring that although the number of points n is large the region is large enough so that the points are sparse in this region. That is, it is rare for a point to have another point close to it. These results extend results for v ? 0 given by Saunders and Funk [9] where it is shown that without the hard core component such results do not hold for v > 0. Statistical applications are discussed.