  Let $S=\Gamma\backslash \mathbb{H}$ be a hyperbolic surface of finite
topological type, such that the Fuchsian group $\Gamma \le
\operatorname{PSL}_2(\mathbb{R})$ is non-elementary, and consider any
generating set $\mathfrak S$ of $\Gamma$. 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 $n\to \infty$, 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. We give a proof of this known
theorem using Gromov's theorem on translation lengths of Gromov-hyperbolic
groups.
