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2007 Asymptotic Distribution of Coordinates on High Dimensional Spheres
Marcus Spruill
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Electron. Commun. Probab. 12: 234-247 (2007). DOI: 10.1214/ECP.v12-1294


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.


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Marcus Spruill. "Asymptotic Distribution of Coordinates on High Dimensional Spheres." Electron. Commun. Probab. 12 234 - 247, 2007.


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

zbMATH: 1132.62012
MathSciNet: MR2335894
Digital Object Identifier: 10.1214/ECP.v12-1294

Primary: 60F17
Secondary: 28A75 , 52A40

Keywords: dependent arrays , empiric distribution , Isoperimetry , micro-canonical ensemble , Minkowski area

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