On the Limit Behavior of a Sequence of Markov Processes Perturbed in a Neighborhood of the Singular Point

Abstract

We study the limit behavior of a sequence of Markov processes whose distributions outside any neighborhood of a “singular” point are attracted to a certain probability law. In any neighborhood of this point, the limit behavior can be irregular. As an example of application of the general result, we consider a symmetric random walk with unit jumps perturbed in the neighborhood of the origin. The invariance principle is established for the standard time and space scaling. The limit process is a skew Brownian motion.