A poisson convergence theorem for a particle system with dependent constant velocities

P. A. Jacobs

Stochastic Processes and their Applications, 1977, vol. 6, issue 1, 41-52

Abstract: Consider an infinite collection of particles travelling in d-dimensional Euclidean space and let Xn denote the initial position of the nth particle. Assume that the nth particle has through all time the random velocity Vn and that {Vn} is a sequence of dependent random variables. Let Xn(t) = Xn + Vnt denote the position of the nth particle at time t. Conditions are obtained for the convergence of {Xn(t)} to a Poisson process as t-->[infinity]. Essentially they require that the dependence in the Vn-sequence decrease with increasing distance between the initial positions and that the conditional distribution of Vn given the initial positions of all the particles and Vn k[not equal to]n be absolutely continuous with respect to Lebesgue measure.

Date: 1977
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