Fluctuations 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

Submitted.

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

Abstract

We study the diameter and perimeter of the convex hull generated by finitely many independent planar random walks whose increments have finite second moments. The large-time fluctuations are governed by the geometry of the polygon formed by the drift vectors. We develop an $L^2$-approximation framework, based on Wald-type maximal central limit theorems, which reduces the asymptotic analysis of the hull to a finite collection of endpoint, maximal-projection, and Brownian support-function terms. For the diameter, we obtain general max-type limit theorems, Gaussian in the case of a unique extremal diametrical pair and typically non-Gaussian when several extremal pairs compete. For the perimeter, we prove a general distributional limit: non-zero extremal drifts contribute maxima of Gaussian projections, while zero-drift extremal walks contribute Brownian support-function terms. The results recover the previously known Gaussian regimes (the case of one or two walks) and identify the non-Gaussian limits in the degenerate and boundary cases left open (even for two walks). We also give $L^2$ approximations of the convex hull by simpler random sets, under Hausdorff and $\ell_1$ metrics on compact convex sets. Our proofs work under the optimal finite second moment assumption.