Position dependent and stochastic thinning of point processes

A short probabilistic proof of Kallenberg's theorem [2] on thinning of point processes is given. It is extended to the case where the probability of deletion of a point depends on the position of the point and is itself random. The proof also leads easily to a statement about the rate of convergence in Renyi's theorem on thinning a renewal process.     

Ordered thinnings of point processes and random measures

This is a study of thinnings of point processes and random measures on the real line that satisfy a weak law of large numbers. The thinning procedures have dependencies based on the order of the points or masses being thinned such that the thinned process is a composition of two random measures. It is shown that the thinned process (normalized by a certain function) converges in distribution if and only if the thinning process does. This result is used to characterize the convergence of thinned processes to infinitely divisible processes, such as a compound Poisson process, when the thinning is independent and nonhomogeneous, stationary, Markovian, or regenerative. Thinning by a sequence of independent identically distributed operations is also discussed. The results here contain Renyi's classical thinning theorem and many of its extensions.     

Inference for thinned point processes, with application to Cox processes

Given i.i.d. point processes N₁, N₂,…, let the observations be p-thinnings N′₁, N′₂,…, where p is a function from the underlying space E (a compact metric space) to [0, 1], whose interpretation is that a point of N_i at x is retained with probability p(x) and deleted with probability 1−p(x). Strongly consistent estimators of the thinning function p and the Laplace functional L_N(f) = E[e^−N(f)] of the N_i are constructed; associated “central limit” properties are given. Tests are presented, for the case when the N_i and N′_i are both observable, of the hypothesis that the N′_i are p-thinnings of the N_i. State estimation techniques are developed for the case where the N_i are Cox processes directed by unobservable random measures M_i; these techniques yield minimum mean-squared error estimators, based on observation of only the thinned processes N′_i of the N_i and the directing measures M_i. Limit theorems for empirical Laplace functionals of point processes are given.     

Constructions for planar point processes using concentric circles

Some point processes are obtained by generalising the well-known construction for a two-dimensional Poisson process which locates an event on each of a sequence of concentric circles in a particular way. The constructions considered here have, in general, a random number of events on each circle. Under certain sufficient conditions, the constructed processes are asymptotically Poisson, far from the origin. The obvious regularity in the structure of these processes can be removed at least superficially, by displacing the events independently off the concentric circles.     

稀疏过程在破产问题中的应用

本讨论一类人寿保险的风险过程，其中保单到达服从齐次Poisson过程。而描述退保及索赔发生的计数过程分别为这一过程的q-稀疏与p-稀疏．对此模型给出其破产概率的具体上界，并与其它一类风险模型进行比较．     

Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain

LetC(A) be the convex hull generated by a Poisson point process in an unbounded convex setA. A representation ofAC(A) as the union of curvilinear triangles with independent areas is established. In the case whenA is a cone the properties of the representation are examined more completely. It is also indicated how to simulateC(A) directly without first simulating the process itself.     

Stochastic order for alpha-permanental point processes

We establish relations of stochastic comparison among point processes elements of the set of alpha-permanental point processes. This set contains in particular, the determinantal point processes, the Poisson point processes and the permanental point processes. We show that these three classes of point processes can be ordered according to the increasing stochastic order. Elementary particles provide illustrations of some of the obtained relations of stochastic comparison.     

稀疏过程在保险公司破产问题中的应用

本文讨论适用于一类人寿保险和财产保险的风险过程 ,其中保单到达服从Poisson过程 ,而描述索赔发生的计数过程为保单到达过程的 p -稀疏过程。对此模型给出了破产概率的上界并对该上界进行了随机模拟 ,同时把所得结果与经典情形进行比较     

索赔为稀疏过程的双复合 Poisson风险模型

研究了一类风险过程,其中保费收入为复合Poisson过程,而描述索赔发生的计数过程为保单到达过程的p-稀疏过程.给出了生存概率满足的积分方程及其在指数分布下的具体表达式,得到了破产概率满足的Lundberg不等式、最终破产概率及有限时间内破产概率的一个上界和生存概率的积分-微分方程,且通过数值例子,分析了初始准备金、保费收入、索赔支付及保单的平均索赔比例对保险公司破产概率的影响.     

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.     

Optimal stochastic control of the intensity of point processes

We consider an intensity control problem for a point process to maximize the expectation of a function of the time when the nth event occurs. We find the optimal control policy when the objective function is unimodal. Moreover, if the objective function is log-concave, so is the value function. As an application, we completely solve an intensity control problem that generalizes the problem studied by Brémaud (1976) and Defourny (2018). Also, we resolve the two conjectures made by Defourny (2018).     

Partitions of point processes: Multivariate poisson approximations

This study shows that when a point process is partitioned into certain uniformly sparse subprocesses, then the subprocesses are asymptotically multivariate Poisson or compound Poisson. Bounds are given for the total-variation distance between the subprocesses and their limits. Several partitioning rules are considered including independent, Markovian, and batch assignments of points.     

Branching random walks,stable point processes and regular variation

Using the theory of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is used to obtain the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. Because of heavy tails of the step size distribution, we can invoke a one large jump principle at the level of point processes to give an explicit representation of the limiting point process. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.     

Statistical inference for detrended point processes

We consider a multivariate point process with a parametric intensity process which splits into a stochastic factor b_t and a trend function a_t of a squared polynomial form with exponents larger than . Such a process occurs in a situation where an underlying process with intensity b_t can be observed on a transformed time scale only. On the basis of the maximum likelihood estimator for the unknown parameter a detrended (or residual) process is defined by transforming the occurrence times via integrated estimated trend function. It is shown that statistics (mean intensity, periodogram estimator) based on the detrended process exhibit the same asymptotic properties as they do in the case of the underlying process (without trend function). Thus trend removal in point processes turns out to be an appropriate method to reveal properties of the (unobservable) underlying process – a concept which is well established in time series. A numerical example of an earthquake aftershock sequence illustrates the performance of the method.     

Optimisation of linear unbiased intensity estimators for point processes

A general non-stationary point process whose intensity function is given up to unknown numerical factor λ is considered. As an alternative to the conventional estimator of λ based on counting the points, we consider general linear unbiased estimators of λ given by sums of weights associated with individual points. A necessary and sufficient condition for a linear, unbiased estimator for the intensity λ to have the minimum variance is determined. It is shown that there are “nearly” no other processes than Poisson and Cox for which the unweighted estimator of λ, which counts the points only, is optimal. The properties of the optimal estimator are illustrated by simulations for the Matérn cluster and the Matérn hard-core processes. This research was partially supported by Grant Agency of Czech Republic, project No. 201/03/D062.     

Perfect simulation and inference for point processes given noisy observations

Summary  The paper is concerned with the exact simulation of an unobserved true point process conditional on a noisy observation. We use dominated coupling from the past (CFTP) on an augmented state space to produce perfect samples of the target marked point process. An optimized coupling of the target chains makes the algorithm considerable faster than with the standard coupling used in dominated CFTP for point processes. The perfect simulations are used for inference and the results are compared to an ordinary Metropolis-Hastings sampler.     

Neural fuzzy point processes

Synaptic events in neural systems were described as generated by an apparatus @ possessing memory and encoding a fuzzy point process (the presynaptic discharge) into another N (the postsynaptic discharge). @ was considered to be a fuzzy automata, for which state membership is dependent on input membership and distribution as well as on a control exercised by other neural structures. In such a device, irregular input distributions favour a direct monotonic codification, whereas regular ones induce discontinuous and inverse relations between both fuzzy point processes. Both behaviors favour analogic and membership relations between the fuzzy input and output. However, there exist intermediate grades of irregularities which result in a context-free encoding, where similitude and equivalence relations predominate. The importance of such findings to neurophysiology is discussed.     

稀疏过程的三特征的联合分布函数

本文考虑一类人寿保险,保费到达为Po isson过程,索赔到达为p-稀疏过程,我们推导三特征的联合分布函数;破产时间,破产概率,破产前的盈余,破产赤字,并由这联合分布得破产概率的显示表达式.