Invariance principle for nonhomogeneous random walks on the grid ℤ1

Abstract

A nonhomogeneous random walk on the grid ℤ1 with transition probabilities that differ from those of a certain homogeneous random walk only at a finite number of points is considered. Trajectories of such a walk are proved to converge to trajectories of a certain generalized diffusion process on the line. This result is a generalization of the well-known invariance principle for the sums of independent random variables and Brownian motion.