Convex hulls of planar random walks with drift

Andrew R. Wade; Chang Xu

doi:10.1090/S0002-9939-2014-12239-8

Convex hulls of planar random walks with drift

Andrew R. Wade^*, Chang Xu

^*Corresponding author for this work

Mathematics And Statistics

Research output: Contribution to journal › Article › peer-review

10 Citations (Scopus)

Abstract

Denote by L_n 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^-1L_n converges almost surely to a deterministic limit and proved an upper bound on the variance Var[L_n] = O(n). We show that n^-1Var[L_n] 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 L_n in the non-degenerate case.

Original language	English
Pages (from-to)	433-445
Number of pages	13
Journal	Proceedings of the American Mathematical Society
Volume	143
Issue number	1
Early online date	16 Sept 2014
DOIs	https://doi.org/10.1090/S0002-9939-2014-12239-8
Publication status	Published - 1 Jan 2015

Keywords

convex hull
random walk
variance asymptotics
central limit theorem

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10.1090/S0002-9939-2014-12239-8

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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.",

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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.

KW - convex hull

KW - random walk

KW - variance asymptotics

KW - central limit theorem

UR - http://www.scopus.com/inward/record.url?scp=84924794863&partnerID=8YFLogxK

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

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EP - 445

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 1

ER -

Convex hulls of planar random walks with drift

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Keywords

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