Convex hulls of random walks

1993/04/01 by Timothy Law Snyder, J. Michael Steele, John Steele · 4 citations
Mathematics · #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics #Limits and Structures in Graph Theory

paper · doi:10.1090/s0002-9939-1993-1169048-2

Abstract

Features related to the perimeter of the convex hull C„ of a random walk in R2 are studied, with particular attention given to its length L„. Bounds on the variance of Ln are obtained to show that, for walks with drift, L„ obeys a strong law. Exponential bounds on the tail probabilities of L„ under special conditions are also obtained. We then develop simple expressions for the expected values of other features of Cn, including the number of faces, the sum of the lengths and squared lengths of the faces, and the number of faces of length t or less.

Citations

• The circumference of a convex polygon
• A Combinatorial Lemma for Complex Numbers

Cited by

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

Related

• On the convex hull of two planar random walks
• Convex hulls of planar random walks
• Convex hulls of planar random walks with drift
• Convex hulls of multidimensional random walks
• Fluctuations for diameter and perimeter of convex hulls of multiple random walks