Limit theorems for radial random walks on homogeneous spaces with growing dimensions

Let X(p) = G(p)/K(p) be homogeneous spaces with compact subgroups K(p) of locally compact groups G(p) with some dimension parameter p such that the double coset spaces G(p)//K(p) can be identified with some fixed locally compact space X. For the projections T(p) : X(p) -> X, and for a given probability measure nu is an element of M(1)(X) there exist unique "radial", i.e. K(p)-invariant measures nu(p) is an element of M(1)(X(p)) with T(p)(nu(p)) = nu as well as associated radial random walks (S(n)(p))(n) on the homogeneous spaces X(p). We generally ask for limit theorems for the random variables T(p) (S(n)(p)) on X for n, p -> infinity. In particular we give a survey about existing results for the Euclidean spaces X(p) = R(p) with K(p) = SO(p) and X = [0, infinity[ as well as to some matrix extension of this rank one setting. Moreover, we derive a new central limit theorem for the hyperbolic spaces X(p) of dimensions p over the skew fields F = R, C, H.