Convex hulls of planar random walks with drift

Andrew R. Wade, Chang Xu

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Denote by Ln the perimeter length of the convex hull of an n-step planar random walk whose increments have finite second moment and nonzero mean. Snyder and Steele showed that n-1Ln converges almost surely to a deterministic limit and proved an upper bound on the variance Var[Ln] = O(n). We show that n-1Var[Ln] converges and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for Ln in the non-degenerate case.

Original languageEnglish
Pages (from-to)433-445
Number of pages13
JournalProceedings of the American Mathematical Society
Volume143
Issue number1
Early online date16 Sep 2014
DOIs
Publication statusPublished - 1 Jan 2015

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Convex Hull
Random walk
Converge
Perimeter
Walk
Central limit theorem
Increment
Upper bound
Denote
Moment
Class

Keywords

  • convex hull
  • random walk
  • variance asymptotics
  • central limit theorem

Cite this

Wade, Andrew R. ; Xu, Chang. / Convex hulls of planar random walks with drift. In: Proceedings of the American Mathematical Society. 2015 ; Vol. 143, No. 1. pp. 433-445.
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Convex hulls of planar random walks with drift. / Wade, Andrew R.; Xu, Chang.

In: Proceedings of the American Mathematical Society, Vol. 143, No. 1, 01.01.2015, p. 433-445.

Research output: Contribution to journalArticle

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AB - Denote by Ln the perimeter length of the convex hull of an n-step planar random walk whose increments have finite second moment and nonzero mean. Snyder and Steele showed that n-1Ln converges almost surely to a deterministic limit and proved an upper bound on the variance Var[Ln] = O(n). We show that n-1Var[Ln] converges and give a simple expression for the limit, which is non-zero for walks outside a certain degenerate class. This answers a question of Snyder and Steele. Furthermore, we prove a central limit theorem for Ln in the non-degenerate case.

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