Approximation of probability measures in variation and products of random matrices

V. N. Tutubalin

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 1, pp. 63-81

Résumé

Let $G$ be the group of real unimodular matrices, $U$ its orthogonal subgroup, $D$ its diagonal subgroup, $g_1,g_2,\dots,g_n,\dots$ a sequence of independent equally distributed random elements of $G$, $g(n)=g_1g_2\dots g_n$. A method is given to approximate the distribution $\mu^n$ of $g(n)$ by a simpler measure $\mu_n$ such that $\operatorname{var}(\mu^n-\widetilde\mu_n)\to0$ as $n\to\infty$. Let $$ g(N)=u_1(n)d(n)u_2(n),\quad u_1(n)\in U,\quad d(n)\in D,\quad u_2(n)\in U $$ Approximations of distributions of $u_1(n)$, $d(n)$ and $u_2(n)$ are given. The joint distribution of these random variables can be approximated as if $u_1(n)$, $d(n)$ and $u_2(n)$ be independent. A conclusion is deduced that the coordinate system $(u_1,d,u_2)$ in $G$ is appropriate to approximate the distribution of $g(n)$. The most general system (coordinates of a matrix are its elements) however appears not to be a good one for this purpose.