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Towards consensus: some convergence theorems on repeated averaging

Published online by Cambridge University Press:  14 July 2016

S. Chatterjee  and
E. Seneta
S. Chatterjee
Affiliation:
New York University
E. Seneta
Affiliation:
Australian National University
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Abstract

The problem of tendency to consensus in an information-exchanging operation is connected with the ergodicity problem for backwards products of stochastic matrices. For such products, weak and strong ergodicity, defined analogously to these concepts for forward products of inhomogeneous Markov chain theory, are shown (in contrast to that theory) to be equivalent. Conditions for ergodicity are derived and their relation to the consensus problem is considered.

Keywords

CONSENSUSREPEATED AVERAGINGSTOCHASTIC MATRICESBACKWARDS PRODUCTSERGODICITYINHOMOGENEOUS MARKOV CHAINS
Type
Research Papers
Information
Journal of Applied Probability , Volume 14 , Issue 1 , March 1977 , pp. 89 - 97
Copyright
Copyright © Applied Probability Trust 1977 

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References

[1] Blackwell, D. (1945) Finite non-homogeneous chains. Ann. Math. 46, 594–599.Google Scholar
[2] Chatterjee, S. (1975) Reaching a consensus; some limit theorems. Proc. Int. Statist. Inst. 159–164.Google Scholar
[3] Cohn, H. (1974) A ratio limit theorem for the finite non-homogeneous Markov chains. Israel J. Math. 19, 329–334.Google Scholar
[4] Dalkey, N. C. (1969) The Delphi method: an experimental study of group opinion. The Rand Corporation, RM-5888.Google Scholar
[5] De Groot, M. H. (1974) Reaching a consensus. J. Amer. Statist. Assoc. 69, 118–121.CrossRefGoogle Scholar
[6] Feller, W. (1968) An Introduction to Probability Theory and its Applications. Vol. 1, 3rd edn. Wiley, New York.Google Scholar
[7] Hajnal, J. (1958) Weak ergodicity in non-homogeneous Markov chains. Proc. Camb. Phil. Soc. 54, 233–246.Google Scholar
[8] Kemeny, J. G. and Snell, L. J. (1960) Finite Markov Chains. Van Nostrand, Princeton, N.J.Google Scholar
[9] Kolmogorov, A. N. (1936) Zur theorie der Markoffschen Ketten. Math. Ann. 112, 155–160.Google Scholar
[10] Mott, J. L. (1957) Conditions for the ergodicity of non-homogeneous finite Markov chains. Proc. Roy. Soc. Edinburgh A 64, 369–380.Google Scholar
[11] Seneta, E. (1973) Non-Negative Matrices. Allen and Unwin, London.Google Scholar
[12] Seneta, E. (1973) On the historical development of the theory of finite inhomogeneous Markov chains. Proc. Camb. Phil. Soc. 74, 507–513.Google Scholar
[13] Winkler, R. L. (1968) The consensus of subjective probability distributions. Management Sci. 15, B61–B75.Google Scholar