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

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