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The convex hull of a planar random walk: perimeter, diameter, and shape
- Source :
- Electronic Journal of Probability, Vol. 23 (2018), article 131
- Publication Year :
- 2018
-
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.<br />Comment: 25 pages
- Subjects :
- Mathematics - Probability
60G50 (Primary) 60D05, 60F05, 60F15, 60F20 (Secondary)
Subjects
Details
- Database :
- arXiv
- Journal :
- Electronic Journal of Probability, Vol. 23 (2018), article 131
- Publication Type :
- Report
- Accession number :
- edsarx.1803.08293
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1214/18-EJP257