Probability densities for interevent time are obtained for a doubly stochastic Poisson point process (DSPP) in the presence of self-excitation. The DSPP is assumed to have a stochastic rate that is a filtered Poisson point process (shot noise). The model of a Poisson process driving another Poisson process produces a pulse-bunching effect. Self-excitation (relative refractoriness) results in a ...

The two-parameter Poisson-Dirichlet distribution is a probability distribution on the totality of positive decreasing sequences with sum 1 and hence considered to govern masses of a random discrete distribution. A characterization of the associated point process (i.e., the random point process obtained by regarding the masses as points in the positive real line) is given in terms of the correla...

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 fo...

We consider a point process obtained summing to each point i of the set of the integer Z an i.i.d random variable ξi having a variance that can be also much larger than 1. We compare the process obtained with this construction with the standard Poisson process, and we show that in some sense our process tends to converge for large variance of ξ to the Poisson process in total variation. We then...

We announce a deterministic analog of Bartlett’s displacement theorem. The result is that a Poisson property is stable with respect to deterministic Hamiltonian displacements. While the random point configurations move according to an n-body evolution, the mean measure P satisfies a nonlinear Vlasov type equation Ṗ + y · ∇xP − ∇y · E(P ) = 0. Combined with Bartlett’s theorem, the result general...

The framework of Stein’s method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem 2.3) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Was...

We show that the random point measures induced by vertices in the convex hull of a Poisson sample on the unit ball, when properly scaled and centered, converge to those of a mean zero Gaussian field. We establish limiting variance and covariance asymptotics in terms of the density of the Poisson sample. Similar results hold for the point measures induced by the maximal points in a Poisson sampl...

The SiZer technique is used to study the homogeneity of a point process of Internet traffic ßow start times. It is seen that a homogenous Poisson process is an inappropriate model, because it does not yield observed statistically signiÞcant burstiness. Some Weibull waiting processes gives better, but still inadequate performance. A clustered Poisson process gives the best Þt.