Some theorems of the strong-law-of-large-numbers type

Let $G$ be the $SL(m)$, $U$ the $S\mathscr O(m)$, $\Gamma$ the diagonal subgroup of $U$ and $X=U/\Gamma$. Consider a sequence $g_1,\dots,g_n,\dots$ of independent identically distributed random elements of $G$. Let
$$ g(n)=g_1g_2\dots g_n=x(n)d(n)u(n), $$
where $x(n)\in X$, $u(n)\in U$ and $d(n)=\operatorname{diag}(e^{t_1(n)},\dots,e^{t_m(n)})$, $t_1(n)<\dots<t_m(n)$. Under some condition on the distribution of $g_i$ the following theorems are proved:
1) there exist real numbers $a_1<a_2<\dots<a_m$ such that, with probability 1,
$$ \frac1nt_k(n)\to a_k,\quad k=1,\dots,m; $$

2) with probability 1, $x(n)\to x(\infty)$, where $x(\infty)$ is a random element of $X$.