2007/01/01 by M. C. Spruill · 2 citations
Mathematics · #Point processes and geometric inequalities #Morphological variations and asymmetry #Stochastic processes and statistical mechanics
paper · pdf · doi:10.1214/ecp.v12-1294
openalex publication_date 2007/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/06
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 √(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.