In this talk, we discuss random walks on isometry groups of Gromov hyperbolic spaces and Teichmüller spaces. We prove that non-elementary random walks exhibit at least linear growth of asymptotic translation lengths without any moment condition. As a corollary, it follows that almost every sample-path on a mapping class group eventually becomes pseudo-Anosov. Moreover, we show that if the underlying measure has a finite first moment, then the growth is linear, as Strong Law of Large Number. This is joint work with Hyungryul Baik and Inhyeok Choi.