The classical law of large numbers (LLN) in probability can be generalized to non-abelian groups to so-called multiplicative ergodic theorems, such as Oseledets’ theorem. We will see that this question can be reformulated in terms of random walks: given a random walk on a geodesic space, do sample paths follow “closely” the geodesics in the space? An analog of LLN would be that the deviation is sublinear in the number of steps.
We will prove a general instance of such a theorem, and see how this applies to the action of the mapping class group on Teichmuller space, proving a conjecture of Kaimanovich. This also has applications to the Poisson boundary.