Limit theorems for radial random walks on homogeneous spaces with growing dimensions
نویسنده
چکیده
Let Xp = Gp/Kp be homogeneous spaces (Kp a compact subgroup of a locally compact group Gp) with dimension parameter p such that the double coset spaces Gp//Kp can be identified with some fixed space X . Then we obtain projections Tp : Xp → X , and for a given probability measure ν ∈ M(X) there exist unique ”radial”, i.e. Kp-invariant measures νp ∈ M1(Xp) with Tp(νp) = ν and associated radial random walks (S n)n on the Xp. We generally ask for limit theorems for Tp(S p n ) for n, p → ∞. In particular we give a survey about existing results for the Euclidean spaces Xp = R p (with Kp = SO(p) and X = [0,∞[)) and matrix spaces, and we derive a new central limit theorem for the hyperbolic spaces Xp of dimensions p over F = R,C,H.
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