Convex hulls of random walks and their scaling limits

Andrew R. Wade, Chang Xu

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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.

Original languageEnglish
Pages (from-to)4300-4320
Number of pages21
JournalStochastic Processes and their Applications
Volume125
Issue number11
Early online date9 Jul 2015
DOIs
Publication statusPublished - Nov 2015

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Keywords

  • Brownian motion
  • convex hull
  • random walk
  • scaling limits
  • variance asymptotics

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