A central limit theorem for isotropic random walks on \(n\)-spheres for \(n\to \infty\)
type: Item
instance of: scholarly article
title: A central limit theorem for isotropic random walks on \(n\)-spheres for \(n\to \infty\)
zbMATH Open document ID: 0822.60063
published in: Journal of Mathematical Analysis and Applications
publication date: 1995-10-05T00:00:00Z
review text: Let \((Y_ l)_{l \geq 0}\) be an isotropic, time-homogeneous Markov chain on \(S^ n\) starting at a fixed point \(x \in S^ n\), satisfying \(P(\cos \angle (Y_ l, x) = r \mid Y_{l-1} = x) =1\) for all \(l\) and \(x\). The following central limit theorem for \(X^ n_ l = \cos \angle (Y_ l, x_ 0)\) is established. Let \(l, n \to \infty\) and \(r \to 1\) in such a manner: \(n^ \rho (1-r) \to 0\), \(\rho > 0\); \(l(1-t) - {1\over 2} \ln n \to b\in (-\infty,\infty]\). Then \(\sqrt {n} X^ n_ l\) converges in distribution to \(N(e^{-b},1)\). The proof is given for a more general setting of Markov chains on \([-1,1]\) with convolution structure by using of limiting properties of ultraspherical polynomials relating to Hermite polynomials and the moment convergence theorem. Further applications of this method of proof are discussed.
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60B15
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 743895
zbMATH Keywords: isotropic random walk
zbMATH Keywords: central limit theorem
zbMATH Keywords: Hermite polynomials
zbMATH Keywords: moment convergence theorem
author: Michael Voit
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1013
OpenAlex ID: W2086417243
DOI: 10.1006/JMAA.1995.1013
description: scientific article; zbMATH DE number 743895