Large Deviations of Poisson Cluster Processes (original) (raw)

Large deviations for fractional Poisson processes

Statistics & Probability Letters, 2013

We prove large deviation principles for two versions of fractional Poisson processes: the main version is a renewal process, the alternative version is a weighted Poisson process. We also present asymptotic results for the ruin probabilities of an insurance model with a fractional Poisson claim number process.

The local principle of large deviations for compound Poisson process with catastrophes

Brazilian Journal of Probability and Statistics, 2021

The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle.

On the number of points of a homogeneous poisson process

Journal of Multivariate Analysis - J MULTIVARIATE ANAL, 1994

We investigate the dense and sparse regions of a d-dimensional Poisson process and establish strong laws for both the maximal and the minimal number of points in families of sets of a certain volume.

Large deviations for a discrete-time marked Hawkes process

arXiv: Probability, 2020

The Hawkes process is a simple point process with wide applications in finance, social networks, criminology, seismology, and many other fields. The Hawkes process is defined for continuous time setting. However, data is often recorded in a discrete-time scheme. In this paper, we study a discrete-time marked Hawkes process, which is appealing for applications. In particular, we study large deviations for the model.

Gamma-Type Results and Other Related Properties of Poisson Processes

Advances in Applied Probability, 1996

Families of Poisson processes defined on general state spaces and with the intensity measure scaled by a positive parameter are investigated. In particular, mean value relations with respect to the scale parameter are established and used to derive various Gamma-type results for certain geometric characteristics determined by finite subprocesses. In particular, we deduce Miles' complementary theorem. Applications of the results within stochastic geometry and particularly for random tessellations are discussed.

Some notes on poisson limits for empirical point processes

Canadian Journal of Statistics, 2009

Define the scaled empirical point process on an independent and identically distributed sequence {Yi : i ≤ n} on IR d as the random point measure with masses at a −1 n Yi. For suitable an we obtain the weak limit of these point processes through a novel use of a dimension-free method based on the convergence of compensators of multiparameter martingales. The method extends previous results in several directions. We obtain limits at points where the density of Yi may be zero, but has regular variation. The joint limit of the empirical process evaluated at distinct points is given by independent Poisson processes. Applications are provided both to nearest-neighbour density estimation in high dimensions, and to the asymptotic behaviour of multivariate extremes such as those arising from bivariate normal copulas.

Size-biased sampling of Poisson point processes and excursions

Probability Theory and Related Fields, 1992

Some general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.

A more comprehensive complementary theorem for the analysis of Poisson point processes

Advances in Applied Probability, 2006

In this paper we discuss the complementary theorem applied to the typicaln-tuple of a Poisson point process. The theorem was first presented by Miles in 1970 and discussed by Santaló in 1976 and, within a Palm measure framework, by Møller and Zuyev in 1996. The theorems put forward by these authors are not correct for all the examples that they present, suggesting that further consideration of their work is needed if one wishes to bring all those examples within the ambit of the complementary theorem. We give alternative analyses of the errant examples and, with a modification of the technicalities in the work of the above authors, move toward a more comprehensive complementary theorem. Some open issues still remain.