Electronic Communications in Probability

Asymptotic Distribution of Coordinates on High Dimensional Spheres

Marcus Spruill

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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
https://projecteuclid.org/euclid.ecp/1465224966

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

Mathematical Reviews number (MathSciNet)
MR2335894

Zentralblatt MATH identifier
1132.62012

Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 52A40: Inequalities and extremum problems 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15]

Keywords
empiric distribution dependent arrays micro-canonical ensemble Minkowski area isoperimetry

Rights
This work is licensed under aCreative 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. https://projecteuclid.org/euclid.ecp/1465224966


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