On the centre of mass of a random walk

Chak Hei Lo and Andrew R. Wade

Stochastic Processes and their Applications, 129, no. 11, November 2019, 4663–4686.

Abstract

For a random walk $S_n$ on $\mathbb{R}^d$ we study the asymptotic behaviour of the associated centre of mass process $G_n = n^{-1} \sum_{i=1}^n S_i$. 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, $G_n$ is recurrent if $d=1$ and transient if $d \geq 2$. In the transient case we show that $G_n$ has diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of heavy-tailed random walks for which $G_n$ is transient in $d=1$.