  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.
