### Abstract

We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere S^{n-1}. In this way, the case of a discretized Brownian motion is related to Gordon's escape theorem dealing with standard Gaussian matrices. We show that for the random walk BM_{n}(i), i ∈ N, the convex hull of the first C^{n} steps (for a sufficiently large universal constant C) contains the origin with probability close to one. Moreover, the approach allows us to prove that with high probability the π/2-covering time of certain random walks on S^{n-1} is of order n. For certain spherical simplices on S^{n-1}, we prove an extension of Gordon's theorem dealing with a broad class of random matrices; as an application, we show that C^{n} steps are sufficient for the standard walk on ℤ^{n} to absorb the origin into its convex hull with a high probability. Finally, we prove that the aforementioned bound is sharp in the following sense: for some universal constant c > 1, the convex hull of the n-dimensional Brownian motion conv(BM_{n}(t): t ∈ [1, c^{n}]) does not contain the origin with probability close to one.

Original language | English (US) |
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Pages (from-to) | 965-1002 |

Number of pages | 38 |

Journal | Annals of Probability |

Volume | 45 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2017 |

### Keywords

- Convex hull
- Covering time
- Random walk

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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## Cite this

^{n}absorb the origin into its convex hull?

*Annals of Probability*,

*45*(2), 965-1002. https://doi.org/10.1214/15-AOP1079