We characterize the possible distributions of a stopped simple symmetric random walk Xτ, where τ is a stopping time relative to the natural filtration of (Xn). We prove that any probability measure on ℤ can be achieved as the law of Xτ where τ is a minimal stopping time, but the set of measures obtained under the further assumption that (X n∧τ : n ≥ 0) is a uniformly integrable martingale is a fractal subset of the set of all centered probability measures on ℤ. This is in sharp contrast to th...