Abstract
On the hyperbolic spaces of the form G/K a fairly complete theory of spherical functions is available in order to study Fourier transforms of K-biinvariant probability measures on G. The differentiability of this Fourier transform enables us to introduce the notion of variance. Moreover, continuous convolution semigroups of probability measures admit a Levy-Khintchine representation, and so Gaussian semi-groups can be defined via their Fourier transforms. The aim of our discussion is to establish sufficient conditions in terms of variances for a triangular system of K-biinvariant probability measures on G to converge towards a Gaussian measure.
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