On multivariate antagonistic marked point processes

This paper investigates two bivariate “antagonistic” processes in the framework of noncooperative games that find applications in economics, engineering, and warfare. These processes represent two players with entirely opposite interests. Hostile actions, associating with one of the components (such as deliberate sellout of stocks of the opponent), take place at random times having a random effect to various vital areas. A consequence of these actions is material damage (such as financial losses) which is associated with the second component. So, in general, each player has two (dependent) ways of striking, while his opponent can sustain multiple damages (of two kinds and interdependent) until his threshold of endurance is reached. Then, the player is ruined. One of the goals is to predict the ruin time of each player as well as the collateral damage to the loser and winner at the ruin time. Furthermore, it is also of vital interest to relate these random elements to the continuous time parameter processes referred to as time sensitive analysis. We succeed in these goals by arriving at closed form joint functionals of the named elements and processes.     

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.     

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.     

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.     

Asymptotic tests of composite hypotheses for non-ergodic type stochastic processes

Limiting distributions of a score statistic and the likelihood ratio statistic for testing a composite hypothesis involving several parameters in non-ergodic type stochastic processes are obtained. It is shown that, unlike in the usual theory (ergodic type processes), the limiting distributions of these statistics are different both under the null and a contiguous sequence of alternative hypotheses. The results are applied to a regression model with explosive autoregressive Gaussian errors. In the discussion of this example a modified score statistic is suggested where the limiting null and non-null distributions are the same as those of the likelihood ratio statistic.     

On large deviations for some sequences of weighted means of Gaussian processes

In this paper we study some sequences of weighted means of continuous real valued Gaussian processes. More precisely we consider suitable generalizations of both arithmetic and logarithmic means of a Gaussian process with covariance function which satisfies either an exponential decay condition or a power decay condition. Our aim is to provide limits of variances of functionals of such weighted means which allow the application of some large deviation results in the literature.     

Quantitative analysis of continuous weighted marked graphs

This paper deals with the computation of the asymptotic firing rate vector and the stationary marking of continuous weighted marked graphs under infinite servers semantics. The continuous weighted marked graphs are a particular class of continuous Petri net where each place has exactly one input and one output transition. First, we give an explicit formula to compute the asymptotic firing rate vector of transitions using the structure of the given net. Then, under the assumption that there exists only one critical circuit in the strongly connected continuous neutral weighted marked graphs, an original approach to compute the vector of stationary marking is presented. Finally, an application to a flexible manufacturing system is given.     

Scaling limits of coupled continuous time random walks and residual order statistics through marked point processes

A continuous time random walk (CTRW) is a random walk in which both spatial changes represented by jumps and waiting times between the jumps are random. The CTRW is coupled if a jump and its preceding or following waiting time are dependent random variables (r.v.), respectively. The aim of this paper is to explain the occurrence of different limit processes for CTRWs with forward- or backward-coupling in Straka and Henry (2011) [37] using marked point processes. We also establish a series representation for the different limits. The methods used also allow us to solve an open problem concerning residual order statistics by LePage (1981) [20].     

Generalized weighted quasi-arithmetic means

Under some natural assumptions on real functions f and g defined on a real interval I, we show that a two variable function M _f,g : I ² → I defined by

Moderate deviations for marked Hawkes processes

We consider a linear Hawkes process with random marks. Some limit theorems have been studied by Karabash and Zhu [Stoch. Models, 31, 433–451 (2015)]. In this paper, we obtain a moderate deviation principle for marked Hawkes processes.     

Inclusion-exclusion and point processes

A combinatorial (inclusion-exclusion) approach to the construction of point processes starting from densities is proposed. A formal sufficient crifficient criterion is derived and then applied with positive results to systems of functions having a special product form. Thus, a new class of point processes is derived to play a role within classical Gibbs processes.     

Time sensitive functionals of marked Cox processes

This paper continues studies on a multivariate marked Cox process C_t observed upon some random epochs τ={τ₁,τ₂,…} initiated in [J. Math. Anal. Appl. 293 (2004) 1-13]. The goal is to connect the continuous time parameter process C_t with C_τ for which closed-form transforms were presented in that paper. This work does not only restore some or much of the information on C_t lost due to its limited observation, but it also manages to scrutinize the behavior of C_τ around the first passage time of C_τ (that takes place upon one of the observation epochs τ) within some random time intervals. Again, analytically tractable formulas for functionals of C_τ are derived.     

On weighted extensions of Cauchy's means

We show that every Cauchy mean in (0,∞) can be embedded into two parameter family of weighted means. Some basic properties and examples are presented. A functional equation which appears in the problem of symmetry of these means is considered. As an application a natural extension of Stolarsky's means is obtained and a two parameter subclass of weighted power means is determined.     

Bayesian modeling of continuously marked spatial point patterns

Many analyses of continuously marked spatial point patterns assume that the density of points, with differing marks, is identical. However, as noted in the seminal paper of Goulard et al. (Scand J Stat 23:365–379, 1996), such an assumption is not realistic in many situations. For example, a stand of forest may have many more small trees than large, hence the model should allow for a higher density of points with small marks. In addition, as suggested by Ogata and Tanemura (Biometrics 41:421–433, 1985), the interaction between points should be a function of their mark, allowing, for example, the range of interaction for large trees to exceed that of smaller trees. The aforementioned articles use frequentist inferential techniques, but interval estimation presents difficulties due to the extremely complex distributional properties of the estimates; it might be possible, however, to use parametric bootstrap methodology for such inferences (Baddeley et al. in J Roy Stat Soc Ser B 67:617–666, 2005). We suggest the use of Bayesian inferential techniques. Although a Bayesian approach requires a complex, computational implementation of (reversible jump) Markov Chain Monte Carlo methodology, it enables a wide variety of inferences (including interval estimation). We demonstrate our approach by analyzing the well known Norway spruce dataset.