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

Abstract

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