Spherical Analysis and Central Limit Theorems on Symmetric Spaces

We prove some results on the kernel of the Abel transform on an irreducible Riemannian symmetric space X = G=K with G noncompact and complex, in particular an estimate of this kernel. We also study the behaviour of spherical functions near the walls of Weyl chambers. We show how these harmonic spherical analysis results lead to a new proof of a central limit theorem of Guivarc'h and Raugi in the complex and K-invariant cases. We present brieey this and other central limit theorems on X. 0. Introduction Let G be a semisimple noncompact Lie group with nite center and K a maximal compact subgroup of G. Harmonic spherical analysis on Riemannian symmetric spaces of the form X = G=K is a very well developed and powerful tool in studying probabilities on such spaces, especially in the central limit problem and the investigating of Gaussian measures. The study of central limit theorems on Riemannian symmetric spaces was initiated by Karpelevich, Shur and Tutubalin 16 and then developed by many authors in several parallel directions. Such theorems nd applications in multivariate statistics and in some engineering problems. 8;22 Interesting results concerning properties of Gaussian measures on symmetric spaces were also obtained (see Section 2).