### 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.

Language | English |
---|---|

Pages | 4300-4320 |

Number of pages | 21 |

Journal | Stochastic Processes and their Applications |

Volume | 125 |

Issue number | 11 |

Early online date | 9 Jul 2015 |

DOIs | |

Publication status | Published - Nov 2015 |

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### Keywords

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

### Cite this

*Stochastic Processes and their Applications*,

*125*(11), 4300-4320. https://doi.org/10.1016/j.spa.2015.06.008

}

*Stochastic Processes and their Applications*, vol. 125, no. 11, pp. 4300-4320. https://doi.org/10.1016/j.spa.2015.06.008

**Convex hulls of random walks and their scaling limits.** / Wade, Andrew R.; Xu, Chang.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convex hulls of random walks and their scaling limits

AU - Wade, Andrew R.

AU - Xu, Chang

PY - 2015/11

Y1 - 2015/11

N2 - 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.

AB - 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.

KW - Brownian motion

KW - convex hull

KW - random walk

KW - scaling limits

KW - variance asymptotics

UR - http://www.scopus.com/inward/record.url?scp=84937140342&partnerID=8YFLogxK

UR - http://www.sciencedirect.com/science/journal/03044149

U2 - 10.1016/j.spa.2015.06.008

DO - 10.1016/j.spa.2015.06.008

M3 - Article

VL - 125

SP - 4300

EP - 4320

JO - Stochastic Processes and their Applications

T2 - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 11

ER -