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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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