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)|
|Number of pages||21|
|Journal||Annales de l'institut Henri Poincare (B) Probability and Statistics|
|State||Published - 2003|
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty