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On laws of large numbers for random walks. (English) Zbl 1111.60005

The authors prove a general law of large numbers for random walks on groups. 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, the authors show that \(\varepsilon\)-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, too.

MSC:

60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60F99 Limit theorems in probability theory
37A30 Ergodic theorems, spectral theory, Markov operators
60J50 Boundary theory for Markov processes
60J65 Brownian motion
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