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

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 considered $N \in \{1,2\}$ and 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.