Random walks and contracting elements I: Deviation inequality and Limit laws

Abstract

We study random walks on metric spaces with contracting isometries. In this first article of the series, we establish sharp deviation inequalities by adapting Gou\"ezel's pivotal time construction. As an application, we establish the exponential bounds for deviation from below, central limit theorem, law of the iterated logarithms and the geodesic tracking of random walks on mapping class groups and CAT(0) spaces.