Growth and recurrence of stationary random walks

Abstract

Let (X n ,n≥1) be a real-valued ergodic stationary stochastic process, and let (Y n =X 1 +…+X n ,n≥1) be the associated random walk. We prove the following: if the sequence of distributions of the random variables Y n /n,n≥1, is uniformly tight (or, more generally, does not have the zero measure as a vague limit point), then there exists a real number c such that the random walk (Y n −nc,n≥1) is recurrent. If this sequence of distributions converges to a probability measure ρ on ℝ (or, more generally, has a nonzero limit ρ in the vague topology), then (Y n −nc,n≥1) is recurrent for ρ−a.e.cℝ.