Invariance Principle for Random Walks and Local Times

Abstract

The classical invariance principle asserts that the distributions of a broad class of continuous functionals of processes constructed from a normalized random walk with finite variance converge to the distributions of these functionals of a Brownian motion. However, nontrivial limit distributions exist also for various discontinuous functionals of random walks, which does not follow directly from the classical invariance principle. Many such functionals of random walks with finite variance have limit distributions expressed in terms of the distributions of functionals of Brownian local time.