Global observables for RW: Law of large numbers

Abstract

We consider the sums TN=∑n=1NF(Sn) where Sn is a random walk on Zd and F:Zd→R is a global observable, that is, a bounded function which admits an average value when averaged over large cubes. We show that TN always satisfies the weak Law of Large Numbers but the strong law fails in general except for one dimensional walks with drift. Under additional regularity assumptions on F, we obtain the Strong Law of Large Numbers and estimate the rate of convergence. The growth exponents which we obtain turn out to be optimal in two special cases: for quasiperiodic observables and for random walks in random scenery.