Limit theorems for random walks on the double coset spaces U(n)∥U(n − 1) for n → ∞

Let (Yjn)j⩾0 be an isotropic random walk on the homogeneous space U(n)U(n − 1) (n⩾2) and πn 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 ⊂ C. Assume the random walk is stopped after j(n) steps. We prove that, under certain restrictions, the random variables (πn(Yj(n)n))n⩾2 on D ⊂ 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.