Global observables for random walks: law of large numbers

Abstract

We consider the sums $T_N={\sum }_{n=1}^{N}F(S_n)$ where $S_n$ is a random walk on ${\mathbb{Z}}^d$ and $F:{\mathbb{Z}}^d\to \mathbb{R}$ is a global observable, that is, a bounded function which admits an average value when averaged over large cubes. We show that $T_N$ 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.

Final version for Annales de l'Institut Henri Poincaré, Probabilités et Statistiques