On the centre of mass of a random walk

TL;DR

This paper investigates the asymptotic behavior of the centre of mass of a random walk in multiple dimensions, establishing recurrence, transience, and rate of escape under various conditions, extending previous work on simple symmetric walks.

Contribution

It provides new conditions for local limit theorems, characterizes recurrence and transience of the centre of mass, and introduces classes of heavy-tailed increments affecting transience.

Findings

$G_n$ is recurrent in 1D with zero mean, finite variance.

$G_n$ is transient in higher dimensions under certain conditions.

Heavy-tailed increments can cause transience even in 1D.

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 random walks with symmetric heavy-tailed increments for which $G_n$ is transient in $d=1$.