On the centre of mass of a random walk

Abstract

For a random walk Sn on Rd we study the asymptotic behaviour of the associated centre of mass process Gn=n−1∑i=1nSi. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, Gn is recurrent if d=1 and transient if d≥2. In the transient case we show that Gn has a diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which Gn is transient in d=1.