 Convex hulls of random walks and their scaling limitsAndrew R. Wade and Chang Xu Stochastic Processes and their Applications, 2015, vol. 125, issue 11, 4300-4320 Abstract: For the perimeter length and the area of the convex hull of the first n steps of a planar random walk, we study n→∞ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random walks with drift (for the area) and walks with no drift (for both area and perimeter length) under mild moments assumptions on the increments. These results complement and contrast with previous work which showed that the perimeter length in the case with drift satisfies a central limit theorem. We deduce these results from weak convergence statements for the convex hulls of random walks to scaling limits defined in terms of convex hulls of certain Brownian motions. We give bounds that confirm that the limiting variances in our results are non-zero. Keywords: Convex hull; Random walk; Brownian motion; Variance asymptotics; Scaling limits (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:125:y:2015:i:11:p:4300-4320 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spa.2015.06.008 Access Statistics for this article Stochastic Processes and their Applications is currently edited by T. Mikosch More articles in Stochastic Processes and their Applications from Elsevier |
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