The central limit theorem for random motions of Euclidean space

Each element $g$ of the group $G$ of all Euclidean motions in $R^3$ can be represented as $g=au$, where $a$ is translation and $u$ is rotation. Consider a sequence $g_1,g_2,\dots, g_n,\dots$ of random independent identically distributed elements of $G$ and their product
$$ g(n)=g_1g_2\dots g_n=a(n)u(n). $$
With natural restrictions the distribution of $\frac1{\sqrt n}a(n)$ tends to a normal distribution as $n\to\infty$, while the distribution of $u(n)$ tends to the normed Haar measure on the group of rotations.