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2018 The convex hull of a planar random walk: perimeter, diameter, and shape
James McRedmond, Andrew R. Wade
Electron. J. Probab. 23: 1-24 (2018). DOI: 10.1214/18-EJP257


We study the convex hull of the first $n$ steps of a planar random walk, and present large-$n$ asymptotic results on its perimeter length $L_n$, diameter $D_n$, and shape. In the case where the walk has a non-zero mean drift, we show that $L_n / D_n \to 2$ a.s., and give distributional limit theorems and variance asymptotics for $D_n$, and in the zero-drift case we show that the convex hull is infinitely often arbitrarily well-approximated in shape by any unit-diameter compact convex set containing the origin, and then $\liminf _{n \to \infty } L_n/D_n =2$ and $\limsup _{n \to \infty } L_n /D_n = \pi $, a.s. Among the tools that we use is a zero-one law for convex hulls of random walks.


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James McRedmond. Andrew R. Wade. "The convex hull of a planar random walk: perimeter, diameter, and shape." Electron. J. Probab. 23 1 - 24, 2018.


Received: 23 March 2018; Accepted: 12 December 2018; Published: 2018
First available in Project Euclid: 22 December 2018

zbMATH: 07021687
MathSciNet: MR3896868
Digital Object Identifier: 10.1214/18-EJP257

Primary: 60G50
Secondary: 60D05 , 60F05 , 60F15 , 60F20

Keywords: Convex hull , diameter , perimeter length , Random walk , shape , Zero-one law


Vol.23 • 2018
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