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

الأصل · EN

The spherical functions of the noncompact Grassmann manifolds Gₚ,q(F)=G/K over the (skew-)fields F=R, C, H with rank q≥1 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 Cqᵇ⊂ Rq of type B. The corresponding product formulas and Harish-Chandra integral representations were recently written down by M. Rösler and the author in an explicit way such that both formulas can be extended analytically to all real parameters p∈[2q-1,∞[, and that associated commutative convolution structures *ₚ on Cqᵇ exist. In this paper we introduce moment functions and the dispersion of probability measures on Cqᵇ 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 (Cqᵇ, *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,F)/U(q,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,∞[.