Empirical distributions in marked point processes

We study the asymptotic behaviour of the empirical distribution function derived from a stationary marked point process when a convex sampling window is expanding without bounds in all directions. We consider a random field model which assumes that the marks and the points are independent and admits dependencies between the marks. The main result is the weak convergence of the empirical process under strong mixing conditions on both independent components of the model. Applying an approximation principle weak convergence can be also shown for appropriately weighted empirical process defined from a stationary d-dimensional germ-grain process with dependent grains.     

On Bernstein type inequalities for stochastic integrals of multivariate point processes

In this paper, we first obtain a Bernstein type of concentration inequality for stochastic integrals of multivariate point processes under some conditions through the Doléans-Dade exponential formula, and then derive a uniform exponential inequality using a generic chaining argument. As a direct consequence, we obtain an upper bound for a sequence of discrete time martingales indexed by a class of functionals. Finally, we apply the uniform exponential bound to nonparametric maximum likelihood estimators and provide a rate of convergence in terms of Hellinger distance, which is an improvement of earlier work of van de Geer (1995).     

Testing separability in marked multidimensional point processes with covariates

In modeling marked point processes, it is convenient to assume a separable or multiplicative form for the conditional intensity, as this assumption typically allows one to estimate each component of the model individually. Tests have been proposed in the simple marked point process case, to investigate whether the mark distribution is separable from the spatial–temporal characteristics of the point process. Here, we extend these tests to the case of a marked point process with covariates, and where one is interested in testing the separability of each of the covariates, as well as the mark and the coordinates of the point process. The extension is not at all trivial, and covariates must be treated in a fundamentally different way than marks and coordinates of the process, especially when the covariates are not uniformly distributed. An application is given to point process models for forecasting wildfire hazard in Los Angeles County, California, and solutions are proposed to the problem of how to proceed when the separability hypothesis is rejected.     

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.     

Weak convergence of non-stationary multivariate marked processes with applications to martingale testing

This paper establishes the weak convergence of a class of marked empirical processes of possibly non-stationary and/or non-ergodic multivariate time series sequences under martingale conditions. The assumptions involved are similar to those in Brown's martingale central limit theorem. In particular, no mixing conditions are imposed. As an application, we propose a test statistic for the martingale hypothesis and we derive its asymptotic null distribution. Finally, a Monte Carlo study shows that the asymptotic results provide good approximations for small and moderate sample sizes. An application to the S&P 500 is also considered.     

Dependence orderings for some functionals of multivariate point processes

We study dependence orderings for functionals of k-variate point processes Φ and Ψ. We view the first process as a collection of counting measures, whereas the second as the sequences of interpoint distances. Subsequently, we establish regularity properties of stationary sequences which generalize known results for iid case. The theoretical results are illustrated by many special cases including comparison of multivariate sums and products, comparison of multivariate shock models and queueing systems.     

Stochastic processes with imbedded marked point processes (pmp) and thcir application in queneing

In generalization of special cases of the literature a class of stochastic processes (PMP) is defined with an imbedded stochastic marked point process of “basic points” which must not be renewal points. A theorem (“intensity conservation principle”) has been proved concerning a relation between stationary distribution of PMP at arbitrary points in time and distributions and intensities connected with the basic points. This relationship simultaneously yields a general method for determination of stationary quantities at arbitrary points in time by means of the corresponding “imbedded” quantities. Some applications to concrete queueing systems have been demonstrated, where arrival or departure epochs of customers are used as basic points. Under weaker independence assumptions as till now done in the literature, new relations are given.     

Rates of convergence for extremes of geometric random variables and marked point processes

We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author’s PhD thesis (Feidt 2013) under the supervision of Andrew D. Barbour.     

Local asymptotic normality of a sequential model for marked point processes and its applications

This paper deals with statistical inference problems for a special type of marked point processes based on the realization in random time intervals [0,_u]. Sufficient conditions to establish the local asymptotic normality (LAN) of the model are presented, and then, certain class of stopping times _u satisfying them is proposed. Using these stopping rules, one can treat the processes within the framework of LAN, which yields asymptotic optimalities of various inference procedures. Applications for compound Poisson processes and continuous time Markov branching processes (CMBP) are discussed. Especially, asymptotically uniformly most powerful tests for criticality of CMBP can be obtained. Such tests do not exist in the case of the non-sequential approach. Also, asymptotic normality of the sequential maximum likelihood estimators (MLE) of the Malthusian parameter of CMBP can be derived, although the non-sequential MLE is not asymptotically normal in the supercritical case.     

On coupling of stochastic processes with embedded point processes

Criteria for semi-, wide-sense-, traditional regeneration and a coupling construction of stochastic processes with embedded point processes are presented.     

On nonparametric ridge estimation for multivariate long-memory processes

Lithuanian Mathematical Journal - We consider nonparametric estimation of the ridge of a probability density function for multivariate linear processes with long-range dependence. We derive...     

On iterative processes generating dense point sets

Summary The central problem of this paper is the question of denseness of those planar point sets <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathcal{P}$, not a subset of a line, which have the property that for every three noncollinear points in $\mathcal{P}$, a specific triangle center (incenter (IC), circumcenter (CC), orthocenter (OC) resp.) is also in the set $\mathcal{P}$. The IC and CC versions were settled before. First we generalize and solve the CC problem in higher dimensions. Then we solve the OC problem in the plane essentially proving that $\mathcal{P}$ is either a dense point set of the plane or it is a subset of a rectangular hyperbola. In the latter case it is either a dense subset or it is a special discrete subset of a rectangular hyperbola.