# Convex hulls of planar random walks with drift

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 language English 433-445 13 Proceedings of the American Mathematical Society 143 1 16 Sep 2014 https://doi.org/10.1090/S0002-9939-2014-12239-8 Published - 1 Jan 2015

### Fingerprint

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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T1 - Convex hulls of planar random walks with drift

AU - Xu, Chang

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N2 - 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.

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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KW - variance asymptotics

KW - central limit theorem

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UR - https://arxiv.org/pdf/1301.4059.pdf

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