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

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.