For planar random walks with finite second moments, the diameter and perimeter of their convex hull converge to Gaussian limits in generic cases and to explicit non-Gaussian limits in degenerate cases, completing the two-walk picture.
Billingsley.Convergence of probability measures
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Fluctuations for diameter and perimeter of convex hulls of multiple random walks
For planar random walks with finite second moments, the diameter and perimeter of their convex hull converge to Gaussian limits in generic cases and to explicit non-Gaussian limits in degenerate cases, completing the two-walk picture.