We prove a general noncommutative law of large numbers. This applies in particular to random walks on any locally finite homogeneous graph, as well as to Brownian motion on Riemannian manifolds which admit a compact quotient. It also generalizes Oseledec’s multiplicative ergodic theorem. In addition, we show that ɛ-shadows of any ballistic random walk with finite moment on any group eventually intersect. Some related results concerning Coxeter groups and mapping class groups are recorded in the last section.
Anders Karlsson. François Ledrappier. "On laws of large numbers for random walks." Ann. Probab. 34 (5) 1693 - 1706, September 2006. https://doi.org/10.1214/009117906000000296