Geodesic
Some theorems of the strong-law-of-large-numbers type
Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 2, pp. 319-326 
V. N. Tutubalin. Some theorems of the strong-law-of-large-numbers type. Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 2, pp. 319-326. http://geodesic.mathdoc.fr/item/TVP_1969_14_2_a12/
@article{TVP_1969_14_2_a12,
     author = {V. N. Tutubalin},
     title = {Some theorems of the strong-law-of-large-numbers type},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {319--326},
     year = {1969},
     volume = {14},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1969_14_2_a12/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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. Under some condition on the distribution of $g_i$ the following theorems are proved: 1) there exist real numbers $a_1 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 $X$.