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Asymptotic Distribution of Coordinates on High Dimensional Spheres

   
@article{ECP1294,
	author = {Marcus Spruill},
	title = {Asymptotic Distribution of Coordinates  on High Dimensional Spheres},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {12},
	year = {2007},
	keywords = {empiric distribution; dependent arrays; micro-canonical ensemble;Minkowski area; isoperimetry},
	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.},
	pages = {no. 23, 234-247},
	issn = {1083-589X},
	doi = {10.1214/ECP.v12-1294},    
        url = {http://ecp.ejpecp.org/article/view/1294}}