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Continuity of Mutual Entropy in the Limiting Signal-To-Noise Ratio Regimes

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Stochastic Analysis 2010

Abstract

This article addresses the issue of the proof of the entropy power inequality (EPI), an important tool in the analysis of Gaussian channels of information transmission, proposed by Shannon. We analyse continuity properties of the mutual entropy of the input and output signals in an additive memoryless channel and discuss assumptions under which the entropy-power inequality holds true.

MSC (2010): 62B10, 94A15

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References

  1. Dobrushin, R.L.: Taking the limit of the argument of entropy and information functions. Theory Probab. Appl. 5(1), 25–32 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley, NY (2006)

    MATH  Google Scholar 

  3. Guo, D., Shamai, S., Verdu, S.: Mutual information and minimum mean-square error in Gaussian channels. IEEE Trans. Inform. Theory 51(4), 1261–1288 (2005)

    Article  MathSciNet  Google Scholar 

  4. Johnson. O., Yu, Y.: Monotonicity, thinning and discrete version of the entropy power inequality, arXiv: 09:09.0541v1

    Google Scholar 

  5. Lieb, E.H.: Proof of entropy conjecture of Wehrl. Commun. Math. Phys. 62(1), 35–41 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  6. Rioul, O.: A simple proof of the entropy-power inequality via properties of mutual information. In: Proceeding of the 2007 IEEE Inter. Symp. on Information Theory. Nice, France (2007)

    Google Scholar 

  7. Shannon, C., Weaver, W.: The mathematical theory of communication. University of Illinois Press, Urbana, IL (1949)

    MATH  Google Scholar 

  8. Stam, A.J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control, 2, 101–112 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  9. Verdu, S., Guo, D.: A simple proof of the entropy–power inequality. IEEE Trans. Inform. Theory 52(5), 2165–2166 (2006)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors acknowledge the support of the Program “Statistical Processes in Communication Science” at Isaac Newton Institute, Cambridge. We would like to express our gratitude to the anonymous referee for his comments, particularly on an incorrect remark in the previous version of the paper.

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Authors and Affiliations

  1. Department of Mathematics, Swansea University, Singleton Park, Swansea, SA2 8PP, United Kingdom

    Mark Kelbert

  2. International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences (RAS), Profsoyuznaya str. 84/32, Moscow, 117997, Russia

    Mark Kelbert

  3. Statistical Laboratory, DPMMS, University of Cambridge, Wilberforce Road, Cambridge, CB 3 0WB, United Kingdom

    Yuri Suhov

  4. Institute for Information Transmission Problems, Russian Academy of Sciences (RAS) 19 Bol’shoi Karetnyi per, Moscow, 101447, Russia

    Yuri Suhov

Authors
  1. Mark Kelbert
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  2. Yuri Suhov
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Correspondence to Mark Kelbert .

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Editors and Affiliations

  1. Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom

    Dan Crisan

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Kelbert, M., Suhov, Y. (2011). Continuity of Mutual Entropy in the Limiting Signal-To-Noise Ratio Regimes. In: Crisan, D. (eds) Stochastic Analysis 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15358-7_14

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