Dispersion and limit theorems for random walks associated with hypergeometric functions of type BC

Abstract

The spherical functions of the noncompact Grassmann manifolds $G_{p,q}(\mathbb F)=G/K$ over the (skew-)fields $\mathbb F=\mathbb R, \mathbb C, \mathbb H$ with rank $q\ge1$ and dimension parameter $p>q$ can be described as Heckman-Opdam hypergeometric functions of type BC, where the double coset space $G//K$ is identified with the Weyl chamber $ C_q^B\subset \mathbb R^q$ of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. R\"osler and the author in an explicit way such that both formulas can be extended analytically to all real parameters $p\in[2q-1,\infty[$, and that associated commutative convolution structures $*_p$ on $C_q^B$ exist. In this paper we introduce moment functions and the dispersion of probability measures on $C_q^B$ depending on $*_p$ and study these functions with the aid of this generalized integral representation. Moreover, we derive strong laws of large numbers and central limit theorems for associated time-homogeneous random walks on $(C_q^B, *_p)$ where the moment functions and the dispersion appear in order to determine drift vectors and covariance matrices of these limit laws explicitely. For integers $p$, all results have interpretations for $G$-invariant random walks on the Grassmannians $G/K$. Besides the BC-cases we also study the spaces $GL(q,\mathbb F)/U(q,\mathbb F)$, which are related to Weyl chambers of type A, and for which corresponding results hold. For the rank-one-case $q=1$, the results of this paper are well-known in the context of Jacobi-type hypergroups on $[0,\infty[$.