Convex hulls of planar random walks with drift

Wade, Andrew R. and Xu, Chang (2015) Convex hulls of planar random walks with drift. Proceedings of the American Mathematical Society, 143 (1). pp. 433-445. ISSN 0002-9939

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Denote by Ln the perimeter length of the convex hull of an n-step planar random walk whose increments have finite second moment and nonzero mean. Snyder and Steele showed that n-1Ln converges almost surely to a deterministic limit and proved an upper bound on the variance Var[Ln] = O(n). We show that n-1Var[Ln] converges and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for Ln in the non-degenerate case.