## Electronic Communications in Probability

### Asymptotic Distribution of Coordinates on High Dimensional Spheres

Marcus Spruill

#### 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.

#### Article information

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

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

https://projecteuclid.org/euclid.ecp/1465224966

Digital Object Identifier
doi:10.1214/ECP.v12-1294

Mathematical Reviews number (MathSciNet)
MR2335894

Zentralblatt MATH identifier
1132.62012

Rights

#### Citation

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

#### References

• Andrews, George E.; Askey, Richard; Roy, Ranjan. Special functions. Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, 1999. xvi+664 pp. ISBN: 0-521-62321-9; 0-521-78988-5
• Barthe, Franck; Guédon, Olivier; Mendelson, Shahar; Naor, Assaf. A probabilistic approach to the geometry of the $l^n_p$-ball. Ann. Probab. 33 (2005), no. 2, 480–513.
• Billingsley, Patrick. Convergence of probability measures. John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp.
• Borel,E. Introduction géométrique á quelques théories physiques. Gauthier-Villars, Paris, 1914.
• Borwein, D.; Borwein, J.; Fee, G.; Girgensohn, R. Refined convexity and special cases of the Blaschke-Santalo inequality. Math. Inequal. Appl. 4 (2001), no. 4, 631–638.
• Busemann, Herbert. Intrinsic area. Ann. of Math. (2) 48, (1947). 234–267.
• Busemann, Herbert. The isoperimetric problem for Minkowski area. Amer. J. Math. 71, (1949). 743–762.
• Busemann, Herbert. A theorem on convex bodies of the Brunn-Minkowski type. Proc. Nat. Acad. Sci. U. S. A. 35, (1949). 27–31.
• Diaconis, Persi; Freedman, David. A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 397–423.
• Friedman, Avner. Foundations of modern analysis. Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London 1970 vi+250 pp.
• Gardner, R. J. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405 (electronic).
• Poincaré,Henri. Calcul des probabilitiés. Gauthier-Villars, Paris, 1912.
• Rachev, S. T.; Rüschendorf, L. Approximate independence of distributions on spheres and their stability properties. Ann. Probab. 19 (1991), no. 3, 1311–1337.
• Schechtman, G.; Zinn, J. On the volume of the intersection of two $Lsp nsb p$ balls. Proc. Amer. Math. Soc. 110 (1990), no. 1, 217–224.
• Serfling, Robert J. Approximation theorems of mathematical statistics. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1980. xiv+371 pp. ISBN: 0-471-02403-1
• Stam, A. J. Limit theorems for uniform distributions on spheres in high-dimensional Euclidean spaces. J. Appl. Probab. 19 (1982), no. 1, 221–228.