Almost Sure Convergence of Uniform Transport Processes to Brownian Motion

Abstract

Let $x_n(t)$ be the position of a particle in one dimension that switches between uniform velocities $+n$ and $-n$ at the jump times of a Poisson process with intensity $n^2$. In this note are constructed realizations of the processes $x_n(t)$ that converge almost surely to Brownian motion, uniformly on the unit time interval.