Non-Gaussian limits for diameter and perimeter of convex hulls of multiple random walks

Wojciech Cygan, Tomislav Kralj, Nikola Sandrić, Stjepan Šebek, Andrew Wade, and Mo Dick Wong

Preprint.

Supported by EPSRC award Anomalous diffusion via self-interaction and reflection (EP/W00657X/1).

Abstract

We prove large-time $L^2$ and distributional limit theorems for perimeter and diameter of the convex hull of~$N$ trajectories of planar random walks whose increments have finite second moments. Earlier work for $N \in \{1,2\}$ showed that, for generic configurations of the mean drifts of the walks, limits are Gaussian. For perimeter, we complete the picture for $N=2$ by showing that the exceptional cases are all non-Gaussian, with limits involving an It\^o integral (two walks with the same non-zero drift) or a geometric functional of Brownian motion (one walk with zero drift and one with non-zero drift), and establish Gaussian limits for generic configurations when $N \geq 3$. For the diameter we obtain a complete picture for $N \geq 2$, with limits (Gaussian or non-Gaussian) described explicitly in terms of the drift configuration. Our approach unifies old and new results in an $L^2$-approximation framework that provides a multivariate extension of Wald's maximal central limit theorem for one-dimensional random walk, and gives certain best-possible approximation results for the convex hull in Hausdorff sense. We also provide variance asymptotics and limiting variances are described explicitly.