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A central limit theorem for semisimple Lie groups

A. D. Virtser

Moscow

Abstract: Let $G$ be a connected semisimple noncompact Lie group with a finite centre, $g_1,g_2,\dots,g_m,\dots$ a sequence of random independent identically distributed elements of $G$ with probability distribution absolutely continuous with respect to the Haar measure on $G$. Denote by $g(m)$ the product $g_1g_2\dotsg_m$.
Asymptotical behavior of the distribution of $g(m)$ as $m\to\infty$ is investigated. The analysis is based on the representation of $g(m)$ in the form $g(m)=x(m)\tilde a(m)k(m)$ where $x(m)$, $k(m)$ are random elements of a maximal compact subgroup of $G$ $\tilde a(m)$ is a random element of an Abelian subgroup of $G$.
It is proved that the factors $x(m)$, $\tilde a(m)$, $k(m)$ are asymptotically independent (theorem 5) and asymptotical behavior of the probability distributions of all the factors is described (theorems 1,2,3,4).

Received: 24.06.1969

 English version: Theory of Probability and its Applications, 1970, 15:4, 667–687

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