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Asymptotic Distribution of Coordinates on High Dimensional Spheres


 
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1. Title Title of document Asymptotic Distribution of Coordinates on High Dimensional Spheres
 
2. Creator Author's name, affiliation, country Marcus C Spruill; Georgia Institute of Technology
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) empiric distribution; dependent arrays; micro-canonical ensemble;Minkowski area; isoperimetry
 
3. Subject Subject classification 60F17; 52A40; 28A75
 
4. Description 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.
 
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7. Date (YYYY-MM-DD) 2007-08-15
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1294
 
10. Identifier Digital Object Identifier 10.1214/ECP.v12-1294
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 12
 
12. Language English=en
 
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