## Electronic Communications in Probability

### Asymptotic Distribution of Coordinates on High Dimensional Spheres

Marcus Spruill

#### Abstract

The coordinates $x_i$ of a point $x = (x_1, x_2, \dots, x_n)$ chosen at random according to a uniform distribution on the $\ell_2(n)$-sphere of radius $n^{1/2}$ have approximately a normal distribution when $n$ is large. The coordinates $x_i$ of points uniformly distributed on the $\ell_1(n)$-sphere of radius $n$ have approximately a double exponential distribution. In these and all the $\ell_p(n),1 \le p \le \infty,$ convergence of the distribution of coordinates as the dimension $n$ increases is at the rate $\sqrt{n}$ and is described precisely in terms of weak convergence of a normalized empirical process to a limiting Gaussian process, the sum of a Brownian bridge and a simple normal process.

#### Article information

Source
Electron. Commun. Probab. Volume 12 (2007), paper no. 23, 234-247.

Dates
Accepted: 15 August 2007
First available in Project Euclid: 6 June 2016

Permanent link to this document
http://projecteuclid.org/euclid.ecp/1465224966

Digital Object Identifier
doi:10.1214/ECP.v12-1294

Mathematical Reviews number (MathSciNet)
MR2335894

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

#### Citation

Spruill, Marcus. Asymptotic Distribution of Coordinates on High Dimensional Spheres. Electron. Commun. Probab. 12 (2007), paper no. 23, 234--247. doi:10.1214/ECP.v12-1294. http://projecteuclid.org/euclid.ecp/1465224966.

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