Convex hulls of planar random walks with drift

2014/09/16 by Andrew R. Wade, Chang Xu · 2 citations
Mathematics · Biochemistry, Genetics and Molecular Biology · #Stochastic processes and statistical mechanics #Diffusion and Search Dynamics #Point processes and geometric inequalities

paper · doi:10.1090/s0002-9939-2014-12239-8

Abstract

Denote by L_n the perimeter length of the convex hull of an n-step planar random walk whose increments have finite second moment and non-zero mean. Snyder and Steele showed that n^-1 L_n converges almost surely to a deterministic limit and proved an upper bound on the variance \mathbb V\mathrm ar [ L_n] = O(n). We show that n^-1 \mathbb V\mathrm ar [L_n] 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 L_n in the non-degenerate case.

Cited by

• Fluctuations for diameter and perimeter of convex hulls of multiple random walks
• Improved bounds on the inradius and inverse inradius process of the convex hull of planar Brownian motion

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