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arXiv · 2509.17590

Fluctuations for diameter and perimeter of convex hulls of multiple random walks

Wojciech Cygan·Tomislav Kralj·Nikola Sandrić·Stjepan Šebek·Andrew Wade·Mo Dick Wong
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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.

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Wojciech Cygan, Tomislav Kralj, Nikola Sandrić, Stjepan Šebek, Andrew Wade, Mo Dick Wong. 2026-07-22. Fluctuations for diameter and perimeter of convex hulls of multiple random walks. https://arxiv.org/abs/2509.17590

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