### 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^{-1}L_{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^{-1}Var[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 |
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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 Sep 2014 |

DOIs | |

Publication status | Published - 1 Jan 2015 |

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### Keywords

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

### Cite this

*Proceedings of the American Mathematical Society*,

*143*(1), 433-445. https://doi.org/10.1090/S0002-9939-2014-12239-8

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*Proceedings of the American Mathematical Society*, vol. 143, no. 1, pp. 433-445. https://doi.org/10.1090/S0002-9939-2014-12239-8

**Convex hulls of planar random walks with drift.** / Wade, Andrew R.; Xu, Chang.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convex hulls of planar random walks with drift

AU - Wade, Andrew R.

AU - Xu, Chang

PY - 2015/1/1

Y1 - 2015/1/1

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

U2 - 10.1090/S0002-9939-2014-12239-8

DO - 10.1090/S0002-9939-2014-12239-8

M3 - Article

AN - SCOPUS:84924794863

VL - 143

SP - 433

EP - 445

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 1

ER -