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An application of the method of moments to the central limit theorem on hyperbolic spaces

Part of the Lecture Notes in Mathematics book series (LNM,volume 861)

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 Lévy-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.

Keywords

  • Symmetric Space
  • Central Limit Theorem
  • Hyperbolic Space
  • Spherical Function
  • Triangular System

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© 1981 Springer-Verlag

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Heyer, H. (1981). An application of the method of moments to the central limit theorem on hyperbolic spaces. In: Dugué, D., Lukacs, E., Rohatgi, V.K. (eds) Analytical Methods in Probability Theory. Lecture Notes in Mathematics, vol 861. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097315

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  • DOI: https://doi.org/10.1007/BFb0097315

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-10823-8

  • Online ISBN: 978-3-540-36785-7

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