Asymptotic behavior of heat kernels on spheres of large dimensions

JOURNAL OF MULTIVARIATE ANALYSIS

卷:59

Article

关键词: Gaussian measures;    ultraspherical polynomials;    hypergroups;    convergence to equilibrium;    total variation distance;    central limit theorem;   

DOI  :  10.1006/jmva.1996.0063

来源: Elsevier

【 摘 要 】

For n greater than or equal to 2, let (mu(tau,n?x))(tau greater than or equal to 0) be the distributions of the Brownian motion on the unit sphere S-n subset of R(n + 1) starting in some point x is an element of S-n. This paper supplements results of Saloff-Coste concerning the rate of convergence of mu(tau, n)(x) to the uniform distribution U-n on S-n for tau --> infinity depending on the dimension n. We show that, lim(n --> infinity) parallel to mu(tau n, n) - U-n parallel to = 2 . erf(e(-s)/root 8) for tau(n) = (1n n + 2s)/(2n), where erf denotes the error function. Our proof depends on approximations of the measures mu(tau, n)(x) by measures which are known explicitly via Poisson kernels on S-n, and which tend, after suitable projections and dilatations, to normal distributions on R for n --> infinity. The above result as well as some further related limit results will be derived in this paper in the slightly more general context of Jacobi-type hypergroups. (C) 1996 Academic Press. Inc.

【 授权许可】

Free