Rate of convergence to Gaussian measures on n-spheres and Jacobi hypergroups

• 成果类型: Article
• 署名作者: Voit, M
• 署名单位: Eberhard Karls University of Tubingen
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1997
• 页码: 457-477
• 关键词: random-walks; convolution; Positivity; series; SPACES

摘要

In this paper we prove central limit theorems of the following kind: let S-d subset of Rd+1 be the unit sphere of dimension d greater than or equal to 2 with uniform distribution omega(d). For each k is an element of N, consider the isotropic random walk (X-n(k))(n greater than or equal to 0) on S-d starting at th north pole with jumps of fixed sizes angle(X-n(k), X-n-1(k)) = pi/root k for all n greater than or equal to 1. Then there is some k(0)(d) such that for all k greater than or equal to k(0)(d), the distributions rho(k) of X-k(k) have continuous, bounded omega(d)-densities f(k). Moreover, there is a (known) Gaussian measure nu on S-d with omega(d)-density h such that \\f(k) = h\\(infinity) = O(1/k) and \\rho(k) - nu\\ = O(1/k) for k --> infinity, where O(1/k) is sharp. We shall derive this rate of convergence in the central limit theorem more generally for a quite general class of isotropic random walks on compact symmetric spaces of rank one as well as for random walks on [0, pi] whose transition probabilities are related to product linearization formulas of Jacobi polynomials.