Abstract
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.
Citation
James McRedmond. Andrew R. Wade. "The convex hull of a planar random walk: perimeter, diameter, and shape." Electron. J. Probab. 23 1 - 24, 2018. https://doi.org/10.1214/18-EJP257
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