On random walk in Lobachevsky's plane

Let $M$ be the Lobachevsky's plane, $G$ its translation group and $mg$ the result of a translation $g\in G$ applied to a point $m\in M$. Consider a sequence $g_1,g_2,\dots,g_n,\dots$ of independent identically distributed random elements of $G$, a point $m_0\in M$ and the distribution $m_0\mu^n$ of the random point $m_0g_1\dots g_n$. Approximations of $m_0\mu^n(A)$ are considered, $A$ being a rather complicated subset of $M$ constructed by means of a discrete subgroup of $G$.