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.
Richard J. Griego. David Heath. Alberto Ruiz-Moncayo. "Almost Sure Convergence of Uniform Transport Processes to Brownian Motion." Ann. Math. Statist. 42 (3) 1129 - 1131, June, 1971. https://doi.org/10.1214/aoms/1177693346