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Groups Complexity Cryptology

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Strong law of large numbers on graphs and groups

Natalia Mosina
  • Department of Mathematics, Engineering, and Computer Science, CUNY/LAGCC, Long Island City, NY, USA.
  • Email:
/ Alexander Ushakov
  • Department of Mathematics, Stevens Institute of Technology, Hoboken, NJ, USA.
  • Email:
Published Online: 2011-02-03 | DOI: https://doi.org/10.1515/gcc.2011.004


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

About the article

Received: 2010-12-22

Published Online: 2011-02-03

Published in Print: 2011-05-01

Citation Information: Groups – Complexity – Cryptology, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc.2011.004. Export Citation

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