A local limit theorem for products of random matrices

V. N. Tutubalin

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 2, pp. 209-221

Résumé

The product $g(n)=g_1\dots g_n$ of random identically distributed independent matrices is represented in the form: $g(n)=x(n)\delta(n)v(n)$, where $x(n)$ and $v(n)$ are unitary matrices, $$ \delta(n)=\operatorname{diag}(\exp\tau_1(n),\dots,\exp\tau_m(n)),\qquad\tau_1(n)<\dots<\tau_m(n). $$ Under conditions 1*) and 2*) the following theorem is proved: The distribution of $g(n)$ can be decomposed into the sum of two measures. The first has the full variation $O(1/n)$. The second is given by the joint density $p_n^*$ of the random variables $$ x(n),\tau^*(n)=\frac{1}{\sqrt n}(\tau(n)-na),v(n), $$ and $$ \sup_{x,t,v}|p_n^*(x,t,v)-\nu_X(x)N_{\sigma^2}(t)\nu_n(v)|\to 0, $$ where $N_{\sigma^2}(t)$ is the normal density on the plane $t_1+\dots+t_m=0$ with non-degenerate, on this plane, variance-covariance matrix $\sigma^2$; $a=(a_1,\dots,a_m)$, $a_1<\dots<a_m$, is а constant vector, and $\nu_x(n)$ and $\nu_n(v)$ are some probability densities on a unitary subgroup ($\nu_n(v)$ is one and the same for all even $n$ and one and the same, may be different, for all odd $n$).