Some central limit theorems for random walks associated with hypergeometric functions of type BC

Abstract

The spherical functions of the noncompact Grassmann manifolds over the real or complex numbers or the quaternions with rank q and dimension parameter p can be seen as Heckman-Opdam hypergeometric functions of type BC, when the double coset space is identified with some Weyl chamber of type B. The associated double coset hypergroups may be embedded into a continuous family of commutative hypergroups with these hypergeometric functions as multiplicative functions with p in some continuous parameter range by a result of R\"osler. Several limit theorems for random walks associated with these hypergroups were recently derived by the second author. We here present further limit theorems in particular for the case where the time parameter as well as p tend to infinity. For integers p, these results admit interpretations for group-invariant random walks on the Grassmann manifolds.