A central limit theorem for isotropic random walks on \(n\)-spheres for \(n\to \infty\)
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.
scientific article; zbMATH DE number 743895