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.
Asymptotic Distribution of Coordinates on High Dimensional Spheres
Published 2007 in Electronic Communications in Probability
ABSTRACT
PUBLICATION RECORD
- Publication year
2007
- Venue
Electronic Communications in Probability
- Publication date
2007-08-15
- Fields of study
Mathematics
- Identifiers
- External record
- Source metadata
Semantic Scholar
CITATION MAP
EXTRACTION MAP
CLAIMS
CONCEPTS
- brownian bridge
A centered Gaussian stochastic process pinned to zero at the endpoints, used as part of the limiting fluctuation process.
- double exponential distribution
A symmetric Laplace-type probability distribution with exponential tails on both sides of its center.
Aliases: Laplace distribution
- normal distribution
The Gaussian probability distribution used here as the limiting form for coordinate values on the \u21132 sphere.
Aliases: Gaussian distribution
- normalized empirical process
A fluctuation process formed from empirical coordinates after centering and scaling to study their asymptotic behavior.
Aliases: empirical process
- simple normal process
A Gaussian process component with normal increments that contributes to the limiting process described in the abstract.
Aliases: normal process
- \u21131(n) sphere
The set of points in R^n constrained by the \u21131 norm to have radius n, with points sampled uniformly from its surface.
Aliases: l1 sphere, \u21131 sphere
- \u21132(n) sphere
The set of points in R^n constrained by the \u21132 norm to have radius n^{1/2}, with points sampled uniformly from its surface.
Aliases: l2 sphere, \u21132 sphere
- \u2113_p(n) sphere family
The collection of high-dimensional spheres defined by the \u2113_p norm for 1 \u2264 p \u2264 \u221e, each with the radius convention used in the abstract.
Aliases: lp sphere family, \u2113_p sphere
- weak convergence
A mode of convergence for probability laws or random processes based on convergence in distribution.
Aliases: convergence in distribution
REFERENCES
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