Abstract: In this seminar we will focus positivity and large deviations type results about the drift on Hyperbolic Spaces. We will build on the results from part one of the seminar to present the naturality of studying large deviations. As for positivity we will relate it with a known result of Furstenberg on the positivity of the Lyapunov exponent. References: Boulanger, A., Mathieu, P., Sert, C., & Sisto, A. (2020). Large deviations for random walks on hyperbolic spaces. arXiv preprint arXiv:2008.02709.