On the centre of mass of a random walk
- Resource Type
- Authors
- Chak Hei Lo; Andrew R. Wade
- Source
- Stochastic processes and their applications, 2019, Vol.129(11), pp.4663-4686 [Peer Reviewed Journal]
- Subject
- Statistics and Probability
Zero mean
Applied Mathematics
Probability (math.PR)
010102 general mathematics
Second moment of area
Random walk
01 natural sciences
Combinatorics
60G50 (Primary) 60F05, 60J10 (Secondary)
010104 statistics & probability
Mathematics::Probability
Modeling and Simulation
Lattice (order)
FOS: Mathematics
0101 mathematics
Mathematics - Probability
Mathematics
- Language
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 $d=1$.
Comment: 26 pages, 1 colour figure; v2: minor revision