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A local limit theorem for products of random matrices
V. N. Tutubalin Moscow
Abstract:
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 à 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
$n$).
Received: 25.02.1976
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