Convex hulls of random walks and their scaling limits

Andrew R. Wade, Chang Xu

Research output: Contribution to journalArticle

5 Citations (Scopus)

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.

LanguageEnglish
Pages4300-4320
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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Brownian movement
Scaling Limit
Convex Hull
Random walk
Perimeter
Asymptotic Variance
Weak Convergence
Walk
Central limit theorem
Increment
Brownian motion
Deduce
Complement
Limiting
Moment

Keywords

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

Cite this

Wade, Andrew R. ; Xu, Chang. / Convex hulls of random walks and their scaling limits. In: Stochastic Processes and their Applications. 2015 ; Vol. 125, No. 11. pp. 4300-4320.
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Convex hulls of random walks and their scaling limits. / Wade, Andrew R.; Xu, Chang.

In: Stochastic Processes and their Applications, Vol. 125, No. 11, 11.2015, p. 4300-4320.

Research output: Contribution to journalArticle

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