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.
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