### Abstract

We prove that the total variation distance between the cone measure and surface measure on the sphere of ℓ _{p} ^{n} is bounded by a constant times 1/ √n. This is used to give a new proof of the fact that the coordinates of a random vector on the ℓ _{p} ^{n} sphere are approximately independent with density proportional to exp(- t ^{p} ), a unification and generalization of two theorems of Diaconis and Freedman. Finally, we show in contrast that a projection of the surface measure of the ℓ _{p} ^{n} sphere onto a random k-dimensional subspace is "close" to the k-dimensional Gaussian measure.

Original language | English (US) |
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Pages (from-to) | 241-261 |

Number of pages | 21 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 39 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2003 |

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

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*Annales de l'institut Henri Poincare (B) Probability and Statistics*,

*39*(2), 241-261. https://doi.org/10.1016/S0246-0203(02)00008-0