Weakly symmetric measures on the group of unimodular matrices (Q1813766)

Let \(G\) be a unimodular (real or complex) group and \(U\) be the orthogonal (unitary) subgroup of the group \(G\). A measure \(\mu\) on the group \(G\) is called weakly symmetric if \(\mu(u^{-1}Au)=\mu(A)\) for all Borel subsets \(A\subseteq G\) and every \(u\in U\). Let \(g_ i\), \(i=1,2,\dots\), be independent (not identically distributed) random elements of the group \(G\) with weakly symmetric distributions and \(g(n)=g_ 1g_ 2\dots g_ n\). \(g(n)\) can be represented as \[ g(n)=k(n)\cdot u(n)\hbox{ and }g(n)=x_ 1(n)\cdot\delta(n)\cdot \gamma(n)\cdot x_ 2(n), \] where \(u(n)\in U\), \(k(n)\) is an upper triangular matrix with positive diagonal elements, \(x_ 1(n)\) and \(x_ 2(n)\) are elements of the group \(G\), the lower angular minors of which are positive, \(\gamma(n)\) is a diagonal matrix of \(U\) and \(\delta(n)\) is a diagonal matrix with positive increasing diagonal elements. Let \((t_ 1(n),\dots,t_ m(n))\) be the vector of the logarithm of the diagonal elements of the matrix \(k(n)\). The author proves the following Theorem. The joint distribution of the matrices \(x_ 1(n)\) and \(x_ 2(n)\) converges when \(n\to \infty\) to the distribution of independent random elements if \(\min_{i=1,\dots,m-1}(t_{i+1}(n)-t_ i(n))\to \infty\).