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Teoriya Veroyatnostei i ee Primeneniya, 1970, Volume 15, Issue 4, Pages 685–705
(Mi tvp1934)
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This article is cited in 24 scientific papers (total in 24 papers)
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
Citation:
A. D. Virtser, “A central limit theorem for semisimple Lie groups”, Teor. Veroyatnost. i Primenen., 15:4 (1970), 685–705; Theory Probab. Appl., 15:4 (1970), 667–687
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