Limit distributions of some integral functionals for null-recurrent diffusions

Abstract

The limit distributions as T→∞ of the functional ∫ 0 T f(X(t)) dt/T are found for one-dimensional null-recurrent Markov processes X( t) under different assumptions concerning local characteristics of X( t) and f. The conditions for the convergence of the distribution of this functional to each distribution in the class of generalized arc-sine law, studied by Lamperti and Watanabe, are obtained.