Limit theorems for quantum modular forms on Fuchsian groups

We discuss the statistical aspects of the quantum modular form, which was introduced by Zagier in terms of a near-modularity property on the set of rational cusps. Examples include classical modular symbols and central values of additive twists rank 2 L-functions. We outline the main ideas for the proof of their limit laws coming from dynamics and hyperbolic geometry (joint with Sandro Bettin and Sary Drappeau).