Limit theorems for random walks on the double coset spaces U(n)//U(n-1) for n->infinity

[摘要] Let (Y-j(n))(j greater than or equal to 0) be an isotropic random walk on the homogeneous space U(n)/U(n - 1) (n greater than or equal to 2) and pi(n) the canonical projection from U(n)/U(n - 1) onto the double coset hypergroup U(n)//U(n - 1) which will be identified with the unit disk D subset of C. Assume the random walk is stopped after j(n) steps. We prove that, under certain restrictions, the random variables (pi(n)(Y-j(n)(n)))(n greater than or equal to 2) on D subset of C admit a central limit theorem. This result has an interpretation related to cut-off phenomena of Diaconis for random walks on hypercubes. The proof depends on a limit relation between the spherical functions of U(n)/U(n - 1) (i.e., certain Jacobi polynomials in two dimensions) and Laguerre polynomials. This limit relation and a connection between Laguerre polynomials and the moments of bivariate normal distributions then assure that the moments of the distributions under consideration tend to the moments of a bivariate normal distribution. The moment convergence criterion will complete the proof.

[发布日期] 1995-12-29

[效力级别]  Proceedings Paper

[关键词] isotropic random walks;symmetric spaces;central limit theorem;moment problem;disk hypergroups;Jacobi polynomials;Laguerre polynomials