  We study a limit behavior of a sequence of Markov processes (or Markov
chains) such that their distributions outside of any neighborhood of a
"singular" point attract to some probability law. In any neighborhood of this
point the behavior may be irregular. As an example of the general result we
consider a symmetric random walk with the unit jump that is perturbed in a
neighborhood of 0. The invariance principle is obtained under standard scaling
of time and space. The limit process turns out to be a skew Brownian motion.
