Thin triangles and a multiplicative ergodic theorem for Teichm\"uller   geometry
  We show that, in the Teichm\"uller metric, "thin-framed triangles are thin"---that is, under suitable hypotheses, the variation of geodesics obeys a hyperbolic-like inequality. This theorem has applications to the study of random walks on Teichm\"uller space. In particular, an application is worked out for the action of the mapping class group: we show that geodesics track random walks sublinearly. 
Moon Duchin