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Convexity and Concentration pp 33–53Cite as

Entropy and Thinning of Discrete Random Variables

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 161))

Abstract

We describe five types of results concerning information and concentration of discrete random variables, and relationships between them, motivated by their counterparts in the continuous case. The results we consider are information theoretic approaches to Poisson approximation, the maximum entropy property of the Poisson distribution, discrete concentration (Poincaré and logarithmic Sobolev) inequalities, monotonicity of entropy and concavity of entropy in the Shepp–Olkin regime.

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References

  1. J. A. Adell, A. Lekuona, and Y. Yu. Sharp bounds on the entropy of the Poisson law and related quantities. IEEE Trans. Inform. Theory, 56(5):2299–2306, May 2010.

    Google Scholar 

  2. S.-i. Amari, O. E. Barndorff-Nielsen, R. E. Kass, S. L. Lauritzen, and C. R. Rao. Differential geometry in statistical inference. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 10. Institute of Mathematical Statistics, Hayward, CA, 1987.

    Google Scholar 

  3. S.-i. Amari and H. Nagaoka. Methods of information geometry, volume 191 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 2000.

    Google Scholar 

  4. L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.

    Google Scholar 

  5. V. Anantharam. Counterexamples to a proposed Stam inequality on finite groups. IEEE Trans. Inform. Theory, 56(4):1825–1827, 2010.

    Article  MathSciNet  Google Scholar 

  6. C. Ané, S. Blachere, D. Chafaï, P. Fougeres, I. Gentil, F. Malrieu, C. Roberto, and G. Scheffer. Sur les inégalités de Sobolev logarithmiques. Panoramas et Syntheses, 10:217, 2000.

    MATH  Google Scholar 

  7. S. Artstein, K. M. Ball, F. Barthe, and A. Naor. On the rate of convergence in the entropic central limit theorem. Probab. Theory Related Fields, 129(3):381–390, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Artstein, K. M. Ball, F. Barthe, and A. Naor. Solution of Shannon’s problem on the monotonicity of entropy. J. Amer. Math. Soc., 17(4):975–982 (electronic), 2004.

    Google Scholar 

  9. D. Bakry and M. Émery. Diffusions hypercontractives. In Séminaire de probabilités, XIX, volume 1123 of Lecture Notes in Math., pages 177–206. Springer, Berlin, 1985.

    Google Scholar 

  10. D. Bakry, I. Gentil, and M. Ledoux. Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der mathematischen Wissenschaften. Springer, 2014.

    Book  MATH  Google Scholar 

  11. K. Ball, F. Barthe, and A. Naor. Entropy jumps in the presence of a spectral gap. Duke Math. J., 119(1):41–63, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Barbour, L. Holst, and S. Janson. Poisson Approximation. Clarendon Press, Oxford, 1992.

    MATH  Google Scholar 

  13. A. Barbour, O. T. Johnson, I. Kontoyiannis, and M. Madiman. Compound Poisson approximation via local information quantities. Electronic Journal of Probability, 15:1344–1369, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  14. A. R. Barron. Entropy and the Central Limit Theorem. Ann. Probab., 14(1):336–342, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  15. J.-D. Benamou and Y. Brenier. A numerical method for the optimal time-continuous mass transport problem and related problems. In Monge Ampère equation: applications to geometry and optimization (Deerfield Beach, FL, 1997), volume 226 of Contemp. Math., pages 1–11. Amer. Math. Soc., Providence, RI, 1999.

    Google Scholar 

  16. J.-D. Benamou and Y. Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math., 84(3):375–393, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  17. N. M. Blachman. The convolution inequality for entropy powers. IEEE Trans. Information Theory, 11:267–271, 1965.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. G. Bobkov, G. P. Chistyakov, and F. Götze. Convergence to stable laws in relative entropy. Journal of Theoretical Probability, 26(3):803–818, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. G. Bobkov, G. P. Chistyakov, and F. Götze. Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem. Ann. Probab., 41(4):2479–2512, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. G. Bobkov, G. P. Chistyakov, and F. Götze. Berry–Esseen bounds in the entropic central limit theorem. Probability Theory and Related Fields, 159(3–4):435–478, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  21. S. G. Bobkov, G. P. Chistyakov, and F. Götze. Fisher information and convergence to stable laws. Bernoulli, 20(3):1620–1646, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. G. Bobkov and M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal., 156(2):347–365, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Borovkov and S. Utev. On an inequality and a related characterisation of the normal distribution. Theory Probab. Appl., 28(2):219–228, 1984.

    Article  MATH  Google Scholar 

  24. P. Brändén. Iterated sequences and the geometry of zeros. J. Reine Angew. Math., 658:115–131, 2011.

    MathSciNet  MATH  Google Scholar 

  25. L. D. Brown. A proof of the Central Limit Theorem motivated by the Cramér-Rao inequality. In G. Kallianpur, P. R. Krishnaiah, and J. K. Ghosh, editors, Statistics and Probability: Essays in Honour of C.R. Rao, pages 141–148. North-Holland, New York, 1982.

    Google Scholar 

  26. T. Cacoullos. On upper and lower bounds for the variance of a function of a random variable. Ann. Probab., 10(3):799–809, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  27. L. A. Caffarelli. Monotonicity properties of optimal transportation and the FKG and related inequalities. Communications in Mathematical Physics, 214(3):547–563, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Caputo, P. Dai Pra, and G. Posta. Convex entropy decay via the Bochner-Bakry-Emery approach. Ann. Inst. Henri Poincaré Probab. Stat., 45(3):734–753, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  29. E. Carlen and A. Soffer. Entropy production by block variable summation and Central Limit Theorems. Comm. Math. Phys., 140(2):339–371, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  30. E. A. Carlen and W. Gangbo. Constrained steepest descent in the 2-Wasserstein metric. Ann. of Math. (2), 157(3):807–846, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  31. D. Chafaï. Binomial-Poisson entropic inequalities and the M/M/∞ queue. ESAIM Probability and Statistics, 10:317–339, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Chernoff. A note on an inequality involving the normal distribution. Ann. Probab., 9(3):533–535, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  33. D. Cordero-Erausquin. Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal., 161(3):257–269, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  34. F. Daly. Negative dependence and stochastic orderings. ESAIM: PS, 20:45–65, 2016. https://doi.org/10.1051/ps/2016002.

    Article  MathSciNet  MATH  Google Scholar 

  35. F. Daly and O. T. Johnson. Bounds on the Poincaré constant under negative dependence. Statistics and Probability Letters, 83:511–518, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  36. A. Dembo, T. M. Cover, and J. A. Thomas. Information theoretic inequalities. IEEE Trans. Information Theory, 37(6):1501–1518, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  37. Y. Derriennic. Entropie, théorèmes limite et marches aléatoires. In H. Heyer, editor, Probability Measures on Groups VIII, Oberwolfach, number 1210 in Lecture Notes in Mathematics, pages 241–284, Berlin, 1985. Springer-Verlag. In French.

    Google Scholar 

  38. M. Erbar and J. Maas. Ricci curvature of finite Markov chains via convexity of the entropy. Archive for Rational Mechanics and Analysis, 206:997–1038, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  39. B. V. Gnedenko and A. N. Kolmogorov. Limit distributions for sums of independent random variables. Addison-Wesley, Cambridge, Mass, 1954.

    MATH  Google Scholar 

  40. B. V. Gnedenko and V. Y. Korolev. Random Summation: Limit Theorems and Applications. CRC Press, Boca Raton, Florida, 1996.

    MATH  Google Scholar 

  41. N. Gozlan, C. Roberto, P.-M. Samson, and P. Tetali. Displacement convexity of entropy and related inequalities on graphs. Probability Theory and Related Fields, 160(1–2):47–94, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  42. L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math., 97(4):1061–1083, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  43. A. Guionnet and B. Zegarlinski. Lectures on logarithmic Sobolev inequalities. In Séminaire de Probabilités, XXXVI, volume 1801 of Lecture Notes in Math., pages 1–134. Springer, Berlin, 2003.

    Google Scholar 

  44. D. Guo, S. Shamai, and S. Verdú. Mutual information and minimum mean-square error in Gaussian channels. IEEE Trans. Inform. Theory, 51(4):1261–1282, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  45. S. Haghighatshoar, E. Abbe, and I. E. Telatar. A new entropy power inequality for integer-valued random variables. IEEE Transactions on Information Theory, 60(7):3787–3796, 2014.

    Article  MathSciNet  Google Scholar 

  46. P. Harremoës. Binomial and Poisson distributions as maximum entropy distributions. IEEE Trans. Information Theory, 47(5):2039–2041, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  47. P. Harremoës, O. T. Johnson, and I. Kontoyiannis. Thinning, entropy and the law of thin numbers. IEEE Trans. Inform. Theory, 56(9):4228–4244, 2010.

    Article  MathSciNet  Google Scholar 

  48. P. Harremoës and C. Vignat. An Entropy Power Inequality for the binomial family. JIPAM. J. Inequal. Pure Appl. Math., 4, 2003. Issue 5, Article 93; see also http://jipam.vu.edu.au/.

  49. E. Hillion. Concavity of entropy along binomial convolutions. Electron. Commun. Probab., 17(4):1–9, 2012.

    MathSciNet  MATH  Google Scholar 

  50. E. Hillion and O. T. Johnson. Discrete versions of the transport equation and the Shepp-Olkin conjecture. Annals of Probability, 44(1):276–306, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  51. E. Hillion and O. T. Johnson. A proof of the Shepp-Olkin entropy concavity conjecture. Bernoulli (to appear), 2017. See also arxiv:1503.01570.

    Google Scholar 

  52. E. Hillion, O. T. Johnson, and Y. Yu. A natural derivative on [0, n] and a binomial Poincaré inequality. ESAIM Probability and Statistics, 16:703–712, 2014.

    Google Scholar 

  53. V. Jog and V. Anantharam. The entropy power inequality and Mrs. Gerber’s Lemma for groups of order 2 n . IEEE Transactions on Information Theory, 60(7):3773–3786, 2014.

    Google Scholar 

  54. O. T. Johnson. Information theory and the Central Limit Theorem. Imperial College Press, London, 2004.

    Book  MATH  Google Scholar 

  55. O. T. Johnson. Log-concavity and the maximum entropy property of the Poisson distribution. Stoch. Proc. Appl., 117(6):791–802, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  56. O. T. Johnson. A de Bruijn identity for symmetric stable laws. In submission, see arXiv:1310.2045, 2013.

    Google Scholar 

  57. O. T. Johnson. A discrete log-Sobolev inequality under a Bakry-Émery type condition. In submission. Ann. L’Inst. Henri Poincaré Probab. Stat. http://imstat.org/aihp/accepted.html.

  58. O. T. Johnson and A. R. Barron. Fisher information inequalities and the Central Limit Theorem. Probability Theory and Related Fields, 129(3):391–409, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  59. O. T. Johnson, I. Kontoyiannis, and M. Madiman. Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete compound Poisson measures. Discrete Applied Mathematics, 161:1232–1250, 2013.

    Article  MathSciNet  MATH  Google Scholar 

  60. O. T. Johnson and Y. Yu. Monotonicity, thinning and discrete versions of the Entropy Power Inequality. IEEE Trans. Inform. Theory, 56(11):5387–5395, 2010.

    Article  MathSciNet  Google Scholar 

  61. I. Johnstone and B. MacGibbon. Une mesure d’information caractérisant la loi de Poisson. In Séminaire de Probabilités, XXI, pages 563–573. Springer, Berlin, 1987.

    Google Scholar 

  62. A. Kagan. A discrete version of the Stam inequality and a characterization of the Poisson distribution. J. Statist. Plann. Inference, 92(1-2):7–12, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  63. J. F. C. Kingman. Uses of exchangeability. Ann. Probability, 6(2):183–197, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  64. C. Klaassen. On an inequality of Chernoff. Ann. Probab., 13(3):966–974, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  65. I. Kontoyiannis, P. Harremoës, and O. T. Johnson. Entropy and the law of small numbers. IEEE Trans. Inform. Theory, 51(2):466–472, 2005.

    Article  MathSciNet  MATH  Google Scholar 

  66. S. Kullback. A lower bound for discrimination information in terms of variation. IEEE Trans. Information Theory, 13:126–127, 1967.

    Article  Google Scholar 

  67. C. Ley and Y. Swan. Stein’s density approach for discrete distributions and information inequalities. See arxiv:1211.3668, 2012.

    Google Scholar 

  68. E. Lieb. Proof of an entropy conjecture of Wehrl. Comm. Math. Phys., 62:35–41, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  69. T. M. Liggett. Ultra logconcave sequences and negative dependence. J. Combin. Theory Ser. A, 79(2):315–325, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  70. Y. Linnik. An information-theoretic proof of the Central Limit Theorem with the Lindeberg Condition. Theory Probab. Appl., 4:288–299, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  71. J. Lott and C. Villani. Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2), 169(3):903–991, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  72. M. Madiman and A. Barron. Generalized entropy power inequalities and monotonicity properties of information. IEEE Trans. Inform. Theory, 53(7):2317–2329, 2007.

    Article  MathSciNet  MATH  Google Scholar 

  73. M. Madiman, J. Melbourne, and P. Xu. Forward and reverse Entropy Power Inequalities in convex geometry. See: arxiv:1604.04225, 2016.

    Google Scholar 

  74. P. Mateev. The entropy of the multinomial distribution. Teor. Verojatnost. i Primenen., 23(1):196–198, 1978.

    MathSciNet  MATH  Google Scholar 

  75. N. Papadatos and V. Papathanasiou. Poisson approximation for a sum of dependent indicators: an alternative approach. Adv. in Appl. Probab., 34(3):609–625, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  76. V. Papathanasiou. Some characteristic properties of the Fisher information matrix via Cacoullos-type inequalities. J. Multivariate Anal., 44(2):256–265, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  77. R. Pemantle. Towards a theory of negative dependence. J. Math. Phys., 41(3):1371–1390, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  78. C. R. Rao. On the distance between two populations. Sankhya, 9:246–248, 1948.

    MathSciNet  Google Scholar 

  79. A. Rényi. A characterization of Poisson processes. Magyar Tud. Akad. Mat. Kutató Int. Közl., 1:519–527, 1956.

    MathSciNet  MATH  Google Scholar 

  80. A. Rényi. On measures of entropy and information. In J. Neyman, editor, Proceedings of the 4th Berkeley Conference on Mathematical Statistics and Probability, pages 547–561, Berkeley, 1961. University of California Press.

    Google Scholar 

  81. B. Roos. Kerstan’s method for compound Poisson approximation. Ann. Probab., 31(4):1754–1771, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  82. M. Shaked and J. G. Shanthikumar. Stochastic orders. Springer Series in Statistics. Springer, New York, 2007.

    Book  MATH  Google Scholar 

  83. S. Shamai and A. Wyner. A binary analog to the entropy-power inequality. IEEE Trans. Inform. Theory, 36(6):1428–1430, Nov 1990.

    Google Scholar 

  84. C. E. Shannon. A mathematical theory of communication. Bell System Tech. J., 27:379–423, 623–656, 1948.

    Google Scholar 

  85. N. Sharma, S. Das, and S. Muthukrishnan. Entropy power inequality for a family of discrete random variables. In 2011 IEEE International Symposium on Information Theory Proceedings (ISIT), pages 1945–1949. IEEE, 2011.

    Google Scholar 

  86. L. A. Shepp and I. Olkin. Entropy of the sum of independent Bernoulli random variables and of the multinomial distribution. In Contributions to probability, pages 201–206. Academic Press, New York, 1981.

    Google Scholar 

  87. R. Shimizu. On Fisher’s amount of information for location family. In Patil, G. P. and Kotz, S. and Ord, J. K., editor, A Modern Course on Statistical Distributions in Scientific Work, Volume 3, pages 305–312. Reidel, 1975.

    Google Scholar 

  88. A. J. Stam. Some inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control, 2:101–112, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  89. K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math., 196(1):65–131, 2006.

    Google Scholar 

  90. K.-T. Sturm. On the geometry of metric measure spaces. II. Acta Math., 196(1):133–177, 2006.

    Google Scholar 

  91. G. Szegő. Orthogonal Polynomials. American Mathematical Society, New York, revised edition, 1958.

    MATH  Google Scholar 

  92. G. Toscani. The fractional Fisher information and the central limit theorem for stable laws. Ricerche di Matematica, 65(1):71–91, 2016.

    Article  MathSciNet  MATH  Google Scholar 

  93. C. Tsallis. Possible generalization of Boltzmann-Gibbs statistics. Journal of Statistical Physics, 52:479–487, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  94. A. Tulino and S. Verdú. Monotonic decrease of the non-Gaussianness of the sum of independent random variables: a simple proof. IEEE Trans. Inform. Theory, 52(9):4295–4297, 2006.

    Article  MathSciNet  MATH  Google Scholar 

  95. C. Villani. Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2003.

    Google Scholar 

  96. C. Villani. Optimal transport: Old and New, volume 338 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 2009.

    Google Scholar 

  97. D. W. Walkup. Pólya sequences, binomial convolution and the union of random sets. J. Appl. Probability, 13(1):76–85, 1976.

    Article  MathSciNet  MATH  Google Scholar 

  98. L. Wang, J. O. Woo, and M. Madiman. A lower bound on the Rényi entropy of convolutions in the integers. In 2014 IEEE International Symposium on Information Theory (ISIT), pages 2829–2833. IEEE, 2014.

    Google Scholar 

  99. H. S. Witsenhausen. Some aspects of convexity useful in information theory. IEEE Trans. Inform. Theory, 26(3):265–271, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  100. L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields, 118(3):427–438, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  101. A. D. Wyner. A theorem on the entropy of certain binary sequences and applications. II. IEEE Trans. Information Theory, 19(6):772–777, 1973.

    Google Scholar 

  102. A. D. Wyner and J. Ziv. A theorem on the entropy of certain binary sequences and applications. I. IEEE Trans. Information Theory, 19(6):769–772, 1973.

    Google Scholar 

  103. Y. Yu. On the maximum entropy properties of the binomial distribution. IEEE Trans. Inform. Theory, 54(7):3351–3353, July 2008.

    Google Scholar 

  104. Y. Yu. Monotonic convergence in an information-theoretic law of small numbers. IEEE Trans. Inform. Theory, 55(12):5412–5422, 2009.

    Article  MathSciNet  Google Scholar 

  105. Y. Yu. On the entropy of compound distributions on nonnegative integers. IEEE Trans. Inform. Theory, 55(8):3645–3650, 2009.

    Article  MathSciNet  Google Scholar 

  106. Y. Yu and O. T. Johnson. Concavity of entropy under thinning. In Proceedings of the 2009 IEEE International Symposium on Information Theory, 28th June - 3rd July 2009, Seoul, pages 144–148, 2009.

    Google Scholar 

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Acknowledgements

The author thanks the Institute for Mathematics and Its Applications for the invitation and funding to speak at the workshop ‘Information Theory and Concentration Phenomena’ in Minneapolis in April 2015. In addition, he would like to thank the organizers and participants of this workshop for stimulating discussions. The author thanks Fraser Daly, Mokshay Madiman and an anonymous referee for helpful comments on earlier drafts of this paper.

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  1. School of Mathematics, University of Bristol, University Walk, BS8 1TW, Bristol, UK

    Oliver Johnson

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  1. Department of Mathematics, Rutgers University, Piscataway, New Jersey, USA

    Eric Carlen

  2. Department of Mathematical Sciences, University of Delaware, Newark, Delaware, USA

    Mokshay Madiman

  3. Department of Mathematics, Case Western Reserve University, Cleveland, Ohio, USA

    Elisabeth M. Werner

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Johnson, O. (2017). Entropy and Thinning of Discrete Random Variables. In: Carlen, E., Madiman, M., Werner, E. (eds) Convexity and Concentration. The IMA Volumes in Mathematics and its Applications, vol 161. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7005-6_2

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