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August, 1983 Random Walks on Discrete Groups: Boundary and Entropy
V. A. Kaimanovich, A. M. Vershik
Ann. Probab. 11(3): 457-490 (August, 1983). DOI: 10.1214/aop/1176993497


The paper is devoted to a study of the exit boundary of random walks on discrete groups and related topics. We give an entropic criterion for triviality of the boundary and prove an analogue of Shannon's theorem for entropy, obtain a boundary triviality criterion in terms of the limit behavior of convolutions and prove a conjecture of Furstenberg about existence of a nondegenerate measure with trivial boundary on any amenable group. We directly connect Kesten's and Folner's amenability criteria by consideration of the spectral measure of the Markov transition operator. Finally we give various examples, some of which disprove some old conjectures.


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V. A. Kaimanovich. A. M. Vershik. "Random Walks on Discrete Groups: Boundary and Entropy." Ann. Probab. 11 (3) 457 - 490, August, 1983.


Published: August, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0641.60009
MathSciNet: MR704539
Digital Object Identifier: 10.1214/aop/1176993497

Primary: 22D40
Secondary: 20F99 , 28D20 , ‎43A07‎ , 60B15 , 60J15 , 60J50

Keywords: $n$-fold convolution , amenability , entropy of random walk , exit boundary , invariant mean , Random walk on group

Rights: Copyright © 1983 Institute of Mathematical Statistics


Vol.11 • No. 3 • August, 1983
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