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Convergence to ends for random walks on the automorphism group of a tree. (English) Zbl 0682.60059
Let \(\mu\) be a regular Borel probability on the group G of automorphisms of a locally finite, infinite tree T and let \(Y_ 1\), \(Y_ 2,..\). denote G-valued i.i.d. r.v.’s with common distribution \(\mu\). Let \(\Omega\) denotes the set of equivalence classes \(\omega\) of sequences \((v_ 1,v_ 2,...)\) of distinct vertices of T, in which \(v_ i\) is a neighbour of \(v_{i+1}\) for each i.
The authors prove that if the support of \(\mu\) is not contained in any amenable subgroup of G then, with probability 1, there exists an end \(\omega\in \Omega\) such that \(Y_ 1Y_ 2,...,Y_ n(v)\to \omega\) in \(T\cup \Omega\) for each fixed \(v\in T\).
Reviewer: R.Norvaiša

MSC:
60G50 Sums of independent random variables; random walks
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60J50 Boundary theory for Markov processes
05C05 Trees
43A05 Measures on groups and semigroups, etc.
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