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Groups Complexity Cryptology

Managing Editor: Rosenberger, Gerhard / Shpilrain, Vladimir

Editorial Board Member: Blackburn, Simon R. / Conder, Marston / Dehornoy, Patrick / Eick, Bettina / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / May, Alexander / Mikhalev, Alexander V. / Myasnikov, Alexei / Roman'kov, Vitalii / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Weil, Pascal

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Mathematical Citation Quotient 2012: 0.25

Strong law of large numbers on graphs and groups

1Department of Mathematics, Engineering, and Computer Science, CUNY/LAGCC, Long Island City, NY, USA.

2Department of Mathematics, Stevens Institute of Technology, Hoboken, NJ, USA.

Citation Information: Groups – Complexity – Cryptology. Volume 3, Issue 1, Pages 67–103, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: 10.1515/gcc.2011.004, February 2011

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We consider (graph-)group-valued random element ξ, discuss the properties of a mean-set 𝔼(ξ), and prove the generalization of the strong law of large numbers for graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev's inequality for ξ and Chernoff-like asymptotic bounds. In addition, we prove several results about configurations of mean-sets in graphs and discuss computational problems together with methods of computing mean-sets in practice and propose an algorithm for such computation.

Keywords.: Probability measures on graphs and groups; average; expectation; mean-set; strong law of large numbers; Chebyshev inequality; Chernoff bound; configuration of mean-sets; free group; shift search problem

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