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

Abstract

In this paper we prove central limit theorems of the following kind: let $S^d \subset \mathbb{R}^{d + 1}$ be the unit sphere of dimension $d \geq 2$ with uniform distribution $\omega_d$. For each $k \epsilon \mathbb{N}$, consider the isotropic random walk $(X_n^k)_{n \geq 0}$ on $S^d$ starting at the north pole with jumps of fixed sizes $\angle (X_n^k, X_{n - 1}^k) = \pi/\sqrt{k}$ for all $n \geq 1$. Then there is some $k_0(d)$ such that for all $k \geq k_0(d)$, the distributions $\varrho_k$ 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 such that $||f_k - h||_{\infty} = O(1/k)$ and $||\varrho_k - \nu|| = O(1/k)$ for $k \to \infty$, 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.