The coincidence approach to stochastic point processes

Abstract

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

Many-body diffusion and path integrals for identical particles

“…This observation has also been used to solve the nodal plane problem for fermions in one dimension [9]. In particle physics the spatial-temporal distribution of bosons and fermions in beams has been obtained using this reduction scheme applied on point processes [38][39][40][41]. For a diffusion process we applied an analogous construction to obtain the propagator of interacting fermions [26].…”

Markov Paths, Loops and Fields

“…For one dimensional diffusions, this point process with independent spacings has clearly an analogue which is the determinantal process with independent spacings (See [51]) defined by the kernel k(x)G(x, y) k(y) (k beeing the killing rate and G the Green function). For one dimensional Brownian motion killed at a constant rate, we recover Macchi point process (Cf [32]). …”

“…For constant killing rate, k(x)G(x, y) k(y) can be expressed as ρ exp(− |x − y| /a), with a, ρ > 0 and 2ρa < 1, the law of the spacings has therefore a density proportional to e − x a sinh( √ 1 − 2ρa x a ) (Cf [32]), which appears to be the convolution of two exponential distributions of parameters of local times of the Poisson process of random loops whose intensity is given by the loop measure defined by the semigroup P t . This will applies to examples related to one-dimensional Brownian motion (or to Markov chains on countable spaces).…”

Standard and robust intensity parameter estimation for stationary determinantal point processes

“…Møller and Waagepetersen (2004) or Illian et al (2008)). First introduced by Macchi (1975), the interesting class of determinantal point processes has been revisited recently by Lavancier et al (2015) in a statistical context. Such processes are in particular designed to model repulsive point patterns.…”