Electronic Communications in Probability

Asymptotic Distribution of Coordinates on High Dimensional Spheres

Marcus Spruill

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

empiric distribution dependent arrays micro-canonical ensemble Minkowski area isoperimetry

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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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