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 Electronic Communications in Probability > Vol. 12 (2007) > Paper 23 open journal systems 


Asymptotic Distribution of Coordinates on High Dimensional Spheres

Marcus C Spruill, Georgia Institute of Technology


Abstract
The coordinates xi of a point x = (x1, x2,..., xn) chosen at random according to a uniform distribution on the l2(n)-sphere of radius n1/2 have approximately a normal distribution when n is large. The coordinates xi of points uniformly distributed on the l1(n)-sphere of radius n have approximately a double exponential distribution. In these and all the lp(n),1 ≤ p ≤ ∞, convergence of the distribution of coordinates as the dimension n increases is at the rate n1/2 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.


Full text: PDF

Pages: 234-247

Published on: August 15, 2007
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Electronic Communications in Probability. ISSN: 1083-589X