Universal Poisson-process limits for general random walks

Set off from a large ensemble of independent and identically distributed random walks, run them for a long time, and take a snapshot of their positions; then, what will you see? To answer this question we take the ensemble-size and the running-time to grow infinitely large, and establish universal Poisson-process limits for the positions. Specifically, we show that the positions of general  linear random walks converge universally to Poisson processes, over  the real line, with uniform and exponential intensities. Also, we show  that the positions of general geometric random walks converge  universally to Poisson processes, over the positive half-line, with  harmonic and power intensities. Examples of the linear random walks include: Brownian motions and Levy motions; motions with Ornstein-Uhlenbeck velocities and with Levy-driven Ornstein-Uhlenbeck velocities; fractional Brownian motions and fractional Levy motions; and Montroll-Weiss random walks. Examples of the geometric random walks include the geometric counterparts of the linear examples. Corollaries to the convergence results further establish that the minimal and maximal positions of the universal Poisson-process limits are governed by the Extreme-Value-Theory statistics: Gumbel in the case of general linear random walks; Weibull and Frechet in the case of general geometric random walks.