Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups
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 th...
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ftculeuclid:oai:CULeuclid:euclid.aop/1024404296 2023-05-15T17:39:55+02:00 Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups Voit, Michael 1997-01 application/pdf http://projecteuclid.org/euclid.aop/1024404296 https://doi.org/10.1214/aop/1024404296 en eng The Institute of Mathematical Statistics 0091-1798 Copyright 1997 Institute of Mathematical Statistics Random walks on $n$-spheres central limit theorem Gaussian measures compact symmetric spaces of rank one total variation distance Jacobi polynomials 60J15 60F05 60B10 33C25 42C10 43A62 Text 1997 ftculeuclid https://doi.org/10.1214/aop/1024404296 2018-10-06T11:34:55Z 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. Text North Pole Project Euclid (Cornell University Library) North Pole The Annals of Probability 25 1 |
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Project Euclid (Cornell University Library) |
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ftculeuclid |
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English |
topic |
Random walks on $n$-spheres central limit theorem Gaussian measures compact symmetric spaces of rank one total variation distance Jacobi polynomials 60J15 60F05 60B10 33C25 42C10 43A62 |
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Random walks on $n$-spheres central limit theorem Gaussian measures compact symmetric spaces of rank one total variation distance Jacobi polynomials 60J15 60F05 60B10 33C25 42C10 43A62 Voit, Michael Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups |
topic_facet |
Random walks on $n$-spheres central limit theorem Gaussian measures compact symmetric spaces of rank one total variation distance Jacobi polynomials 60J15 60F05 60B10 33C25 42C10 43A62 |
description |
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. |
format |
Text |
author |
Voit, Michael |
author_facet |
Voit, Michael |
author_sort |
Voit, Michael |
title |
Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups |
title_short |
Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups |
title_full |
Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups |
title_fullStr |
Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups |
title_full_unstemmed |
Rate of convergence to Gaussian measures on $n$-spheres and Jacobi hypergroups |
title_sort |
rate of convergence to gaussian measures on $n$-spheres and jacobi hypergroups |
publisher |
The Institute of Mathematical Statistics |
publishDate |
1997 |
url |
http://projecteuclid.org/euclid.aop/1024404296 https://doi.org/10.1214/aop/1024404296 |
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North Pole |
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North Pole |
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North Pole |
genre_facet |
North Pole |
op_relation |
0091-1798 |
op_rights |
Copyright 1997 Institute of Mathematical Statistics |
op_doi |
https://doi.org/10.1214/aop/1024404296 |
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The Annals of Probability |
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25 |
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1 |
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1766140679059144704 |