Let $G$ be the group of isometries of the Poincaré disk. Let $\mu$ be a probability measure on $G$, and consider $g_1,..,g_n$ i.i.d. random variables on $G$ distributed according to $\mu$. For $z\in \mathbb{C}, |z|<1$, I would like to understand what can be said about the sequence

$$ z_n = g_n...g_1 (z),\quad z\in \mathbb{C}, |z|<1. $$

In particular, I wish to get an understanding of:

-when does $d(0,z_n)$ goes to infinity

-when it does, what is the behavior of $z_n$ as it gets closer to the boundary (does it converges?)

I feel that it is a standard question but I couldn't find yet a result which corresponds precisely to this setting. Could anyone help? Thank you.

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