Probability laws for the distribution of geometric lengths when sampling by a random walk in a Fuchsian fundamental group

Let $S=Gammabackslash mathbb{H}$ be a hyperbolic surface of finite topological type, such that the Fuchsian group $Gamma le operatorname{PSL}_2(mathbb{R})$ is non-elementary. We prove that there exists a generating set $mathfrak S$ of $Gamma$ satisfying the following: when sampling by an $n$-step random walk in $pi_1(S) cong Gamma$ with each step given by an element in $mathfrak S$, the subset of this sampled set comprised of hyperbolic elements approaches full measure as $nto infty$, and for this subset, the distribution of geometric lengths obeys a Law of Large Numbers, Central Limit Theorem, Large Deviations Principle, and Local Limit Theorem.