  We study the quenched behaviour of a perturbed version of the simple
symmetric random walk on the set of integers. The random walker moves
symmetrically with an exception of some randomly chosen sites where we impose a
random drift. We show that if the gaps between the marked sites are i.i.d. and
regularly varying with a sufficiently small index, then there is no strong
quenched limit laws for the position of the random walker. As a consequence we
study the quenched limit laws in the context of weak convergence of random
measures.

Probability