Convex hulls of random walks and their scaling limits

Andrew R. Wade and Chang Xu

Stochastic Processes and their Applications, 125, no. 11, November 2015, 4300–4320. DOI: 10.1016/j.spa.2015.06.008.

Supported by EPSRC award Non-homogeneous random walks (EP/J021784/1).

Abstract

For the perimeter length and the area of the convex hull of the first $n$ steps of a planar random walk, we study large $n$ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random walks with drift (for the area) and walks with no drift (for both area and perimeter length) under mild moments assumptions on the increments. These results complement and contrast with previous work which showed that the perimeter length in the case with drift satisfies a central limit theorem. We deduce these results from weak convergence statements for the convex hulls of random walks to scaling limits defined in terms of convex hulls of certain Brownian motions. We give bounds that confirm that the limiting variances in our results are non-zero.

Further remarks

Here are some pictures of convex hulls of random walks.