{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T06:49:14Z","timestamp":1774680554955,"version":"3.50.1"},"reference-count":177,"publisher":"Emerald","issue":"1-2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2013,10,23]]},"abstract":"<jats:p>During the last two decades, concentration inequalities have been the subject of exciting developments in various areas, including convex geometry, functional analysis, statistical physics, high-dimensional statistics, pure and applied probability theory (e.g., concentration of measure phenomena in random graphs, random matrices, and percolation), information theory, theoretical computer science, and learning theory. This monograph focuses on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding. In addition to being a survey, this monograph also includes various new recent results derived by the authors. This monograph focuses on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding. In addition to being a survey, this monograph also includes various new recent results derived by the authors.<\/jats:p>\n                  <jats:p>The first part of the monograph introduces classical concentration inequalities for martingales, as well as some recent refinements and extensions. The power and versatility of the martingale approach is exemplified in the context of codes defined on graphs and iterative decoding algorithms, as well as codes for wireless communication.<\/jats:p>\n                  <jats:p>The second part of the monograph introduces the entropy method, an information-theoretic technique for deriving concentration inequalities. The basic ingredients of the entropy method are discussed first in the context of logarithmic Sobolev inequalities, which underlie the so-called functional approach to concentration of measure, and then from a complementary information-theoretic viewpoint based on transportation-cost inequalities and probability in metric spaces. Some representative results on concentration for dependent random variables are briefly summarized, with emphasis on their connections to the entropy method. Finally, we discuss several applications of the entropy method to problems in communications and coding, including strong converses, empirical distributions of good channel codes, and an information-theoretic converse for concentration of measure.<\/jats:p>","DOI":"10.1561\/0100000064","type":"journal-article","created":{"date-parts":[[2013,10,23]],"date-time":"2013-10-23T08:36:19Z","timestamp":1382517379000},"page":"1-247","source":"Crossref","is-referenced-by-count":103,"title":["Concentration of Measure Inequalities in Information Theory, Communications, and Coding"],"prefix":"10.1561","volume":"10","author":[{"given":"Maxim","family":"Raginsky","sequence":"first","affiliation":[{"name":"Department of Electrical and Computer Engineering, Coordinated Science Laboratory, University of Illinois at Urbana-Champaign Urbana , IL 61801,","place":["USA"]}]},{"given":"Igal","family":"Sason","sequence":"additional","affiliation":[{"name":"Department of Electrical Engineering, Technion \u2013 Israel Institute of Technology , Haifa 32000,","place":["Israel"]}]}],"member":"140","published-online":{"date-parts":[[2013,10,23]]},"reference":[{"issue":"1","key":"2026032712265640800_ref001","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/aop\/1042644705","article-title":"A new look at independence","volume":"24","author":"Talagrand","year":"1996","journal-title":"Annals of Probability"},{"key":"2026032712265640800_ref002","doi-asserted-by":"crossref","DOI":"10.1093\/acprof:oso\/9780199535255.001.0001","volume-title":"Concentration Inequalities - A Nonasymptotic Theory of Independence","author":"Boucheron","year":"2013"},{"key":"2026032712265640800_ref003","volume-title":"The Concentration of Measure Phenomenon, volume 89 of Mathematical Surveys and Monographs","author":"Ledoux","year":"2001"},{"key":"2026032712265640800_ref004","unstructured":"G.\n              Lugosi\n            \n          . Concentration of measure inequalities - lecture notes, 2009. URL: http:\/\/www.econ.upf.edu\/~lugosi\/anu.pdf."},{"key":"2026032712265640800_ref005","volume-title":"The Concentration of Measure Phenomenon, volume 1896 of Lecture Notes in Mathematics","author":"Massart","year":"2007"},{"key":"2026032712265640800_ref006","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1007\/978-3-662-12788-9_6","volume-title":"Probabilistic Methods for Algorithmic Discrete Mathematics","author":"McDiarmid","year":"1998"},{"key":"2026032712265640800_ref007","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BF02699376","article-title":"Concentration of measure and isoperimteric inequalities in product space","volume":"81","author":"Talagrand","year":"1995","journal-title":"Publications Math\u00e9matiques de l\u2019I.H.E.S"},{"key":"2026032712265640800_ref008","doi-asserted-by":"crossref","first-page":"357","DOI":"10.2748\/tmj\/1178243286","article-title":"Weighted sums of certain dependent random variables","volume":"19","author":"Azuma","year":"1967","journal-title":"Tohoku Mathematical Journal"},{"issue":"301","key":"2026032712265640800_ref009","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1080\/01621459.1963.10500830","article-title":"Probability inequalities for sums of bounded random variables","volume":"58","author":"Hoeffding","year":"1963","journal-title":"Journal of the American Statistical Association"},{"key":"2026032712265640800_ref010","article-title":"Spencer","volume-title":"The Probabilistic Method","author":"Alon","year":"2008","edition":"third"},{"key":"2026032712265640800_ref011","volume-title":"Complex Graphs and Networks, volume 107 of Regional Conference Series in Mathematics","author":"Chung","year":"2006"},{"issue":"1","key":"2026032712265640800_ref012","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1080\/15427951.2006.10129115","article-title":"Concentration inequalities and martingale inequalities: a survey","volume":"3","author":"Chung","year":"2006","journal-title":"Internet Mathematics"},{"key":"2026032712265640800_ref013","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511791338","volume-title":"Modern Coding Theory","author":"Richardson","year":"2008"},{"issue":"12","key":"2026032712265640800_ref014","doi-asserted-by":"crossref","first-page":"7086","DOI":"10.1109\/TIT.2012.2211334","article-title":"PAC-Bayesian inequalities for martingales","volume":"58","author":"Seldin","year":"2012","journal-title":"IEEE Trans. on Information Theory"},{"issue":"4","key":"2026032712265640800_ref015","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/s10208-011-9099-z","article-title":"User-friendly tail bounds for sums of random matrices","volume":"12","author":"Tropp","year":"2012","journal-title":"Foundations of Computational Mathematics"},{"key":"2026032712265640800_ref016","doi-asserted-by":"crossref","first-page":"262","DOI":"10.1214\/ECP.v16-1624","article-title":"Freedman\u2019s inequality for matrix martingales","volume":"16","author":"Tropp","year":"2011","journal-title":"Electronic Communications in Probability"},{"issue":"4","key":"2026032712265640800_ref017","first-page":"635","article-title":"Transport inequalities: a survey","volume":"16","author":"Gozlan","year":"2010","journal-title":"Markov Processes and Related Fields"},{"key":"2026032712265640800_ref018","volume-title":"Probability Theory and Combinatorial Optimization, volume 69 of CBMS\u2013NSF Regional Conference Series in Applied Mathematics","author":"Steele","year":"1997"},{"issue":"2","key":"2026032712265640800_ref019","doi-asserted-by":"crossref","first-page":"927","DOI":"10.1214\/aop\/1024404424","article-title":"Information inequalities and concentration of measure","volume":"25","author":"Dembo","year":"1997","journal-title":"Annals of Probability"},{"key":"2026032712265640800_ref020","volume-title":"Concentration Inequalities with Exchangeable Pairs","author":"Chatterjee","year":"2008"},{"key":"2026032712265640800_ref021","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1007\/s00440-006-0029-y","article-title":"Stein\u2019s method for concentration inequalities","volume":"138","author":"Chatterjee","year":"2007","journal-title":"Probability Theory and Related Fields"},{"issue":"6","key":"2026032712265640800_ref022","doi-asserted-by":"crossref","first-page":"2443","DOI":"10.1214\/10-AOP542","article-title":"Applications of Stein\u2019s method for concentration inequalities","volume":"38","author":"Chatterjee","year":"2010","journal-title":"Annals of Probability"},{"key":"2026032712265640800_ref023","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1214\/11-PS182","article-title":"Fundamentals of Stein\u2019s method","volume":"8","author":"Ross","year":"2011","journal-title":"Probability Surveys"},{"key":"2026032712265640800_ref024","unstructured":"E.\n              Abbe\n             and A.Montanari. On the concentration of the number of solutions of random satisfiability formulas, 2010. URL: http:\/\/arxiv.org\/abs\/1006.3786."},{"key":"2026032712265640800_ref025","first-page":"2801","article-title":"On the concentration of the capacity for a code division multiple access system","volume-title":"Proceedings of the 2007 IEEE International Symposium on Information Theory","author":"Korada","year":"2007"},{"key":"2026032712265640800_ref026","first-page":"1433","article-title":"Concentration of magnetization for linear block codes","volume-title":"Proceedings of the 2008 IEEE International Symposium on Information Theory","author":"Korada","year":"2008"},{"key":"2026032712265640800_ref027","unstructured":"S.\n              Kudekar\n            \n          . Statistical Physics Methods for Sparse Graph Codes. PhD thesis, EPFL - Swiss Federal Institute of Technology, Lausanne, Switzeland, July2009. URL: http:\/\/infoscience.epfl.ch\/record\/138478\/files\/EPFL_TH4442.pdf."},{"issue":"10","key":"2026032712265640800_ref028","doi-asserted-by":"crossref","first-page":"4635","DOI":"10.1109\/TIT.2009.2027523","article-title":"Sharp bounds for optimal decoding of low-density parity-check codes","volume":"55","author":"Kudekar","year":"2009","journal-title":"IEEE Trans. on Information Theory"},{"issue":"11","key":"2026032712265640800_ref029","doi-asserted-by":"crossref","first-page":"5590","DOI":"10.1109\/TIT.2010.2070131","article-title":"Tight bounds on the capacity of binary input random CDMA systems","volume":"56","author":"Korada","year":"2010","journal-title":"IEEE Trans. on Information Theory"},{"issue":"9","key":"2026032712265640800_ref030","doi-asserted-by":"crossref","first-page":"3247","DOI":"10.1109\/TIT.2005.853320","article-title":"Tight bounds for LDPC and LDGM codes under MAP decoding","volume":"51","author":"Montanari","year":"2005","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref031","volume-title":"Mean Field Models for Spin Glasses","author":"Talagrand","year":"2010"},{"issue":"4","key":"2026032712265640800_ref032","doi-asserted-by":"crossref","first-page":"1528","DOI":"10.1214\/10-AOP592","article-title":"Concentration of the information in data with log-concave distributions","volume":"39","author":"Bobkov","year":"2011","journal-title":"Annals of Probability"},{"issue":"8","key":"2026032712265640800_ref033","doi-asserted-by":"crossref","first-page":"4940","DOI":"10.1109\/TIT.2011.2158475","article-title":"The entropy per coordinate of a random vector is highly constrained under convexity conditions","volume":"57","author":"Bobkov","year":"2011","journal-title":"IEEE Trans. on Information Theory"},{"issue":"1","key":"2026032712265640800_ref034","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/BF02579208","article-title":"Sharp concentration of the chromatic number on random graphs","volume":"7","author":"Shamir","year":"1987","journal-title":"Combinatorica"},{"issue":"2","key":"2026032712265640800_ref035","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1109\/18.910575","article-title":"Efficient erasure-correcting codes","volume":"47","author":"Luby","year":"2001","journal-title":"IEEE Trans. on Information Theory"},{"issue":"2","key":"2026032712265640800_ref036","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1109\/18.910577","article-title":"The capacity of low-density paritycheck codes under message-passing decoding","volume":"47","author":"Richardson","year":"2001","journal-title":"IEEE Trans. on Information Theory"},{"issue":"6","key":"2026032712265640800_ref037","doi-asserted-by":"crossref","first-page":"1710","DOI":"10.1109\/18.556667","article-title":"Expander codes","volume":"42","author":"Sipser","year":"1996","journal-title":"IEEE Trans. on Information Theory"},{"issue":"6","key":"2026032712265640800_ref038","doi-asserted-by":"crossref","first-page":"3207","DOI":"10.1109\/TIT.2011.2137210","article-title":"Probability estimation in the rare-events regime","volume":"57","author":"Wagner","year":"2011","journal-title":"IEEE Trans. on Information Theory"},{"issue":"1","key":"2026032712265640800_ref039","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1017\/S0963548396002854","article-title":"Centering sequences with bounded differences","volume":"6","author":"McDiarmid","year":"1997","journal-title":"Combinatorics, Probability and Computing"},{"key":"2026032712265640800_ref040","first-page":"32","article-title":"On the random coding exponent of nonlinear Gaussian channels","volume-title":"Proceedings of the 2009 IEEE International Workshop on Information Theory","author":"Xenoulis","year":"2009"},{"issue":"3","key":"2026032712265640800_ref041","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.1109\/TIT.2011.2104571","article-title":"Achievable rates for nonlinear Volterra channels","volume":"57","author":"Xenoulis","year":"2011","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref042","first-page":"1430","article-title":"New achievable rates for nonlinear Volterra channels via martingale inequalities","volume-title":"Proceedings of the 2012 IEEE International Workshop on Information Theory","author":"Xenoulis","year":"2012"},{"key":"2026032712265640800_ref043","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1051\/ps:1997103","article-title":"On Talagrand\u2019s deviation inequalities for product measures","volume":"1","author":"Ledoux","year":"1997","journal-title":"ESAIM: Probability and Statistics"},{"issue":"4","key":"2026032712265640800_ref044","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.2307\/2373688","article-title":"Logarithmic Sobolev inequalities","volume":"97","author":"Gross","year":"1975","journal-title":"American Journal of Mathematics"},{"key":"2026032712265640800_ref045","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/S0019-9958(59)90348-1","article-title":"Some inequalities satisfied by the quantities of information of Fisher and Shannon","volume":"2","author":"Stam","year":"1959","journal-title":"Information and Control"},{"issue":"1","key":"2026032712265640800_ref046","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1063\/1.1664760","article-title":"A partially alternate derivation of a result of Nelson","volume":"10","author":"Federbush","year":"1969","journal-title":"Journal of Mathematical Physics"},{"issue":"6","key":"2026032712265640800_ref047","doi-asserted-by":"crossref","first-page":"1501","DOI":"10.1109\/18.104312","article-title":"Information theoretic inequalities","volume":"37","author":"Dembo","year":"1991","journal-title":"IEEE Trans. on Information Theory"},{"issue":"4","key":"2026032712265640800_ref048","doi-asserted-by":"crossref","first-page":"1695","DOI":"10.1109\/18.850718","article-title":"A short proof of the \u2018concavity of entropy power\u2019","volume":"46","author":"Villani","year":"2000","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref049","article-title":"An information-theoretic proof of Nash\u2019s inequality","volume-title":"Rendiconti Lincei: Matematica e Applicazioni","author":"Toscani","year":"2012"},{"key":"2026032712265640800_ref050","first-page":"1","article-title":"Lectures on logarithmic Sobolev inequalities","volume":"36","author":"Guionnet","year":"2002","journal-title":"S\u00e9minaire de probabilit\u00e9s (Strasbourg)"},{"key":"2026032712265640800_ref051","first-page":"120","volume-title":"S\u00e9minaire de Probabilit\u00e9s XXXIII, volume 1709 of Lecture Notes in Math","author":"Ledoux","year":"1999"},{"key":"2026032712265640800_ref052","volume-title":"An Invitation to Logarithmic Sobolev Inequalities, volume 14 of SFM\/AMS Texts and Monographs","author":"Royer","year":"2007"},{"key":"2026032712265640800_ref053","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1006\/jfan.1998.3326","article-title":"Exponential integrability and transportation cost related to logarithmic Sobolev inequalities","volume":"163","author":"Bobkov","year":"1999","journal-title":"Journal of Functional Analysis"},{"issue":"2","key":"2026032712265640800_ref054","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1006\/jfan.1997.3187","article-title":"On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures","volume":"156","author":"Bobkov","year":"1998","journal-title":"Journal of Functional Analysis"},{"issue":"2","key":"2026032712265640800_ref055","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1007\/s10959-006-0016-3","article-title":"Modified logarithmic Sobolev inequalities in discrete settings","volume":"19","author":"Bobkov","year":"2006","journal-title":"Journal of Theoretical Probability"},{"issue":"2","key":"2026032712265640800_ref056","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1215\/kjm\/1250283556","article-title":"Entropies, convexity, and functional inequalities: \u03a6-entropies and \u03a6-Sobolev inequalities","volume":"44","author":"Chafa\u00ef","year":"2004","journal-title":"J. Math. Kyoto University"},{"issue":"6","key":"2026032712265640800_ref057","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1109\/TIT.2009.2018179","article-title":"Logarithmic Sobolev inequalities for information measures","volume":"55","author":"Kitsos","year":"2009","journal-title":"IEEE Trans. on Information Theory"},{"issue":"2","key":"2026032712265640800_ref058","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1214\/aop\/1039639365","article-title":"Bounding d-distance by informational divergence: a method to prove measure concentration","volume":"24","author":"Marton","year":"1996","journal-title":"Annals of Probability"},{"key":"2026032712265640800_ref059","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/058","volume-title":"Topics in Optimal Transportation","author":"Villani","year":"2003"},{"key":"2026032712265640800_ref060","volume-title":"Optimal Transport: Old and New","author":"Villani","year":"2008"},{"key":"2026032712265640800_ref061","doi-asserted-by":"crossref","first-page":"342","DOI":"10.1016\/j.matpur.2006.06.003","article-title":"On quadratic transportation cost inequalities","volume":"86","author":"Cattiaux","year":"2006","journal-title":"Journal de Mat\u00e9matiques Pures et Appliqu\u00e9es"},{"key":"2026032712265640800_ref062","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1214\/ECP.v1-979","article-title":"Transportation approach to some concentration inequalities in product spaces","volume":"1","author":"Dembo","year":"1996","journal-title":"Electronic Communications in Probability"},{"issue":"3B","key":"2026032712265640800_ref063","doi-asserted-by":"crossref","first-page":"2702","DOI":"10.1214\/009117904000000531","article-title":"Transportation cost-information inequalities and applications to random dynamical systems and diffusions","volume":"32","author":"Djellout","year":"2004","journal-title":"Annals of Probability"},{"issue":"6","key":"2026032712265640800_ref064","doi-asserted-by":"crossref","first-page":"2480","DOI":"10.1214\/09-AOP470","article-title":"A characterization of dimension free concentration in terms of transportation inequalities","volume":"37","author":"Gozlan","year":"2009","journal-title":"Annals of Probability"},{"key":"2026032712265640800_ref065","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/s00440-010-0328-1","article-title":"Properties of isoperimetric, functional and transportentropy inequalities via concentration","volume":"152","author":"Milman","year":"2012","journal-title":"Probability Theory and Related Fields"},{"issue":"2","key":"2026032712265640800_ref066","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1214\/aop\/1176996402","article-title":"A generalization of Ornstein\u2019s d distance with applications to information theory","volume":"3","author":"Gray","year":"1975","journal-title":"Annals of Probability"},{"issue":"5","key":"2026032712265640800_ref067","doi-asserted-by":"crossref","first-page":"524","DOI":"10.1109\/TIT.1975.1055440","article-title":"Process definitions of distortion-rate functions and source coding theorems","volume":"21","author":"Gray","year":"1975","journal-title":"IEEE Trans. on Information Theory"},{"issue":"1","key":"2026032712265640800_ref068","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1109\/18.481779","article-title":"Simulation of random processes and ratedistortion theory","volume":"42","author":"Steinberg","year":"1996","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref069","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1007\/BF00535682","article-title":"Bounds on conditional probabilities with applications in multi-user communication","volume":"34","author":"Ahlswede","year":"1976","journal-title":"Z. Wahrschein-lichkeitstheorie verw. Gebiete"},{"key":"2026032712265640800_ref070","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1007\/BF00535683","article-title":"Every bad code has a good subcode: a local converse to the coding theorem","volume":"34","author":"Ahlswede","year":"1976","journal-title":"Z. Wahrscheinlichkeitstheorie verw. Gebiete"},{"issue":"3","key":"2026032712265640800_ref071","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1109\/TIT.1986.1057176","article-title":"A simple proof of the blowing-up lemma","volume":"32","author":"Marton","year":"1986","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref072","unstructured":"Y.\n              Altu\u02c7g\n             and A. B.Wagner. Refinement of the sphere-packing bound: asymmetric channels, 2012. URL: http:\/\/arxiv.org\/abs\/1211.6997."},{"issue":"2","key":"2026032712265640800_ref073","doi-asserted-by":"crossref","first-page":"473","DOI":"10.1109\/TIT.2008.2009580","article-title":"Finite-length scaling for iteratively decoded LDPC ensembles","volume":"55","author":"Amraoui","year":"2009","journal-title":"IEEE Trans. on Information Theory"},{"issue":"7","key":"2026032712265640800_ref074","doi-asserted-by":"crossref","first-page":"4770","DOI":"10.1109\/TIT.2012.2191668","article-title":"Analytical solution of covariance evolution for irregular LDPC codes","volume":"58","author":"Nozaki","year":"2012","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref075","unstructured":"I.\n              Kontoyiannis\n             and S.Verd\u00fa. Lossless data compression at finite blocklengths, 2012. URL: http:\/\/arxiv.org\/abs\/1212.2668."},{"issue":"6","key":"2026032712265640800_ref076","doi-asserted-by":"crossref","first-page":"3309","DOI":"10.1109\/TIT.2012.2186786","article-title":"Fixed-length lossy compression in the finite blocklength regime","volume":"58","author":"Kostina","year":"2012","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref077","doi-asserted-by":"crossref","first-page":"7036","DOI":"10.1109\/TIT.2012.2210695","article-title":"A linear program for the finite block length converse of Polyanskiy-Poor-Verd\u00fa via nonsignaling codes","volume":"(12)","author":"Matthews","year":"2012","journal-title":"IEEE Trans. on Information Theory"},{"issue":"5","key":"2026032712265640800_ref078","doi-asserted-by":"crossref","first-page":"2307","DOI":"10.1109\/TIT.2010.2043769","article-title":"Channel coding rate in finite blocklength regime","volume":"56","author":"Polyanskiy","year":"2010","journal-title":"IEEE Trans. on Information Theory"},{"issue":"5","key":"2026032712265640800_ref079","doi-asserted-by":"crossref","first-page":"1962","DOI":"10.1109\/TIT.2008.920216","article-title":"An improved sphere-packing bound for finite-length codes on symmetric channels","volume":"54","author":"Wiechman","year":"2008","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref080","doi-asserted-by":"crossref","DOI":"10.1142\/6300","volume-title":"A First Look at Rigorous Probability Theory","author":"Rosenthal","year":"2006","edition":"second"},{"key":"2026032712265640800_ref081","volume-title":"Large Deviations Techniques and Applications","author":"Dembo","year":"1997","edition":"second"},{"issue":"4","key":"2026032712265640800_ref082","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1214\/aoms\/1177729330","article-title":"A measure of asymptotic efficiency of tests of a hypothesis based on the sum of observations","volume":"23","author":"Chernoff","year":"1952","journal-title":"Annals of Mathematical Statistics"},{"key":"2026032712265640800_ref083","volume-title":"The Theory of Probability","author":"Bernstein","year":"1927"},{"key":"2026032712265640800_ref084","volume-title":"Multiuser Detection","author":"Verd\u00fa","year":"1998"},{"key":"2026032712265640800_ref085","first-page":"148","volume-title":"Surveys in Combinatorics","author":"McDiarmid","year":"1989"},{"key":"2026032712265640800_ref086","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-2545-2","volume-title":"Weak Convergence and Empirical Processes","author":"van der Vaart","year":"1996"},{"key":"2026032712265640800_ref087","first-page":"311","article-title":"Large deviation methods for approximate probabilistic inference","author":"Kearns"},{"issue":"3","key":"2026032712265640800_ref088","first-page":"1","article-title":"On the concentration of the missing mass","volume":"18","author":"Berend","year":"2013","journal-title":"Electronic Communications in Probability"},{"key":"2026032712265640800_ref089","first-page":"1","article-title":"A refinement of Hoeffding\u2019s inequality","volume-title":"Journal of Statistical Computation and Simulation","author":"From","year":"2011"},{"key":"2026032712265640800_ref090","volume-title":"Probability and Measure","author":"Billingsley","year":"1995","edition":"3rd"},{"key":"2026032712265640800_ref091","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198572237.001.0001","volume-title":"Probability and Random Processes","author":"Grimmett","year":"2001","edition":"third"},{"key":"2026032712265640800_ref092","first-page":"1563","article-title":"Relative entropy and exponential deviation bounds for general Markov chains","author":"Kontoyiannis"},{"issue":"9","key":"2026032712265640800_ref093","doi-asserted-by":"crossref","first-page":"2568","DOI":"10.1109\/TIT.2002.800480","article-title":"Random codes: minimum distances and error exponents","volume":"48","author":"Barg","year":"2002","journal-title":"IEEE Trans. on Information Theory"},{"issue":"8","key":"2026032712265640800_ref094","doi-asserted-by":"crossref","first-page":"1692","DOI":"10.1109\/TIT.2004.831763","article-title":"A logarithmic upper bound on the minimum distance of turbo codes","volume":"50","author":"Breiling","year":"2004","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref095","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/4347.001.0001","volume-title":"Low-Density Parity-Check Codes","author":"Gallager","year":"1963"},{"issue":"6","key":"2026032712265640800_ref096","doi-asserted-by":"crossref","first-page":"2173","DOI":"10.1109\/18.782170","article-title":"Which codes have cycle-free Tanner graphs?","volume":"45","author":"Etzion","year":"1999","journal-title":"IEEE Trans. on Information Theory"},{"issue":"7","key":"2026032712265640800_ref097","doi-asserted-by":"crossref","first-page":"2956","DOI":"10.1109\/TIT.2009.2021305","article-title":"On universal properties of capacity-approaching LDPC code ensembles","volume":"55","author":"Sason","year":"2009","journal-title":"IEEE Trans. on Information Theory"},{"issue":"2","key":"2026032712265640800_ref098","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1109\/18.910576","article-title":"Improved low-density parity-check codes using irregular graphs","volume":"47","author":"Luby","year":"2001","journal-title":"IEEE Trans. on Information Theory"},{"issue":"7","key":"2026032712265640800_ref099","doi-asserted-by":"crossref","first-page":"1636","DOI":"10.1109\/TIT.2003.813563","article-title":"Binary intersymbol interference channels: Gallager bounds, density evolution, and code performance bounds","volume":"49","author":"Kav\u010di\u0107","year":"2003","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref100","article-title":"Aspects of Convex Optimization and Concentration in Coding","volume-title":"MSc thesis, Department of Electrical Engineering, Technion - Israel Institute of Technology, Haifa, Israel","author":"Eshel","year":"2012"},{"issue":"1","key":"2026032712265640800_ref101","doi-asserted-by":"crossref","first-page":"507","DOI":"10.1002\/ett.4460060506","article-title":"Iterative correction of intersymbol interference: turboequalization","volume":"6","author":"Douillard","year":"1995","journal-title":"European Transactions on Telecommunications"},{"issue":"12","key":"2026032712265640800_ref102","doi-asserted-by":"crossref","first-page":"5277","DOI":"10.1109\/TIT.2008.2006466","article-title":"Maxwell construction: the hidden bridge between iterative and maximum apposteriori decoding","volume":"54","author":"M\u00e9asson","year":"2008","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref103","first-page":"153","article-title":"Capacity-achieving sequences","volume":"123","author":"Shokrollahi","year":"2000","journal-title":"Volume in Mathematics and its Applications"},{"key":"2026032712265640800_ref104","volume-title":"Wireless Communications","author":"Molisch","year":"2005"},{"key":"2026032712265640800_ref105","unstructured":"G.\n              Wunder\n            , R. F. H.Fischer, H.Boche, S.Litsyn, and J. S.No. The PAPR problem in OFDM transmission: new directions for a long-lasting problem. accepted to the IEEE Signal Processing Magazine, December2012. [Online]. Available: http:\/\/arxiv.org\/abs\/1212.2865."},{"issue":"3","key":"2026032712265640800_ref106","doi-asserted-by":"crossref","first-page":"992","DOI":"10.1109\/TIT.2005.864433","article-title":"Generalized bounds on the crest-factor istribution of OFDM signals with applications to code design","volume":"52","author":"Litsyn","year":"2006","journal-title":"IEEE Trans. on Information Theory"},{"issue":"1","key":"2026032712265640800_ref107","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/BF02393433","article-title":"Some properties of trigonometric series whose terms have random signs","volume":"91","author":"Salem","year":"1954","journal-title":"Acta Mathematica"},{"issue":"2","key":"2026032712265640800_ref108","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1109\/TIT.2002.807311","article-title":"New results on the statistical distribution of the crest-factor of OFDM signals","volume":"49","author":"Wunder","year":"2003","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref109","first-page":"784","article-title":"On the concentration of the crest factor for OFDM signals","author":"Sason","year":"2011"},{"key":"2026032712265640800_ref110","volume-title":"Principles of Digital Transmission with Wireless Applications","author":"Benedetto","year":"1999"},{"issue":"8","key":"2026032712265640800_ref111","doi-asserted-by":"crossref","first-page":"1928","DOI":"10.1016\/j.spl.2013.04.015","article-title":"Tightened exponential bounds for discrete-time conditionally symmetric martingales with bounded jumps","volume":"83","author":"Sason","year":"2013","journal-title":"Statistics and Probability Letters"},{"key":"2026032712265640800_ref112","unstructured":"X.\n              Fan\n            , I.Grama, and Q.Liu. Hoeffding\u2019s inequality for supermartingales, 2011. URL: http:\/\/arxiv.org\/abs\/1109.4359."},{"key":"2026032712265640800_ref113","unstructured":"X.\n              Fan\n            , I.Grama, and Q.Liu. The missing factor in Bennett\u2019s inequality, 2012. URL: http:\/\/arxiv.org\/abs\/1206.2592."},{"key":"2026032712265640800_ref114","doi-asserted-by":"crossref","DOI":"10.1561\/9781933019796","volume-title":"Performance Analysis of Linear Codes under Maximum-Likelihood Decoding: A Tutorial, volume 3 of Foundations and Trends in Communications and Information Theory","author":"Sason","year":"2006"},{"issue":"335-395","key":"2026032712265640800_ref115","article-title":"Ultracontractivity and the heat kernel for Schr\u00f6dinger operators and Dirichlet Laplacians","volume":"59","author":"Davies","year":"1984","journal-title":"Journal of Functional Analysis"},{"issue":"11","key":"2026032712265640800_ref116","doi-asserted-by":"crossref","first-page":"5030","DOI":"10.1109\/TIT.2008.929968","article-title":"The information lost in erasures","volume":"54","author":"Verd\u00fa","year":"2008","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref117","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/0022-1236(91)90155-X","article-title":"Superadditivity of Fisher\u2019s information and logarithmic Sobolev inequalities","volume":"101","author":"Carlen","year":"1991","journal-title":"Journal of Functional Analysis"},{"issue":"6","key":"2026032712265640800_ref118","doi-asserted-by":"crossref","first-page":"1265","DOI":"10.2307\/2374139","article-title":"Gross\u2019s logarithmic Sobolev inequality: a simple proof","volume":"101","author":"Adams","year":"1979","journal-title":"American Journal of Mathematics"},{"key":"2026032712265640800_ref119","volume-title":"London Mathematical Society Lecture Notes","author":"Blower","year":"2009"},{"key":"2026032712265640800_ref120","doi-asserted-by":"crossref","DOI":"10.1142\/p341","volume-title":"Information Theory and the Central Limit Theorem","author":"Johnson","year":"2004"},{"key":"2026032712265640800_ref121","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/014","volume-title":"Analysis","author":"Lieb","year":"2001","edition":"2nd"},{"issue":"6","key":"2026032712265640800_ref122","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1109\/TIT.1984.1056983","article-title":"On the similarity of the entropy power inequality and the Brunn\u2013Minkowski inequality","volume":"30","author":"Costa","year":"1984","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref123","doi-asserted-by":"crossref","DOI":"10.1002\/9780470434697","volume-title":"Robust Statistics","author":"Huber","year":"2009","edition":"second"},{"key":"2026032712265640800_ref124","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/s00440-004-0344-0","article-title":"Fisher information inequalities and the central limit theorem","volume":"129","author":"Johnson","year":"2004","journal-title":"Probability Theory and Related Fields"},{"issue":"8","key":"2026032712265640800_ref125","doi-asserted-by":"crossref","first-page":"3712","DOI":"10.1109\/TIT.2010.2050800","article-title":"Mismatched estimation and relative entropy","volume":"56","author":"Verd\u00fa","year":"2010","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref126","volume-title":"Detection, Estimation and Modulation Theory, Part I","author":"van Trees","year":"1968"},{"key":"2026032712265640800_ref127","volume-title":"Measure Theory and Fine Properties of Functions","author":"Evans","year":"1992"},{"key":"2026032712265640800_ref128","volume-title":"Time\u2019s Arrow: The Origins of Thermodynamic Behavior","author":"Mackey","year":"1992"},{"key":"2026032712265640800_ref129","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03620-4","volume-title":"Stochastic Differential Equations: An Introduction with Applications","author":"\u00d8ksendal","year":"1998","edition":"5"},{"key":"2026032712265640800_ref130","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4684-0302-2","volume-title":"Brownian Motion and Stochastic Calculus","author":"Karatzas","year":"1988","edition":"second"},{"key":"2026032712265640800_ref131","doi-asserted-by":"crossref","DOI":"10.1142\/p386","volume-title":"Introduction to Stochastic Calculus with Applications","author":"Klebaner","year":"2005","edition":"second"},{"key":"2026032712265640800_ref132","unstructured":"T.\n              van Erven\n             and P.Harremo\u0451s. R\u00e9nyi divergence and Kullback\u2013Leibler divergence. IEEE Trans. on Information Theory, 2012. submitted, 2012. URL: http:\/\/arxiv.org\/abs\/1206.2459."},{"key":"2026032712265640800_ref133","volume-title":"Elements of Information Theory","author":"Cover","year":"2006","edition":"second"},{"issue":"2","key":"2026032712265640800_ref134","doi-asserted-by":"crossref","first-page":"434","DOI":"10.3150\/10-BEJ341","article-title":"Thermodynamics and concentration","volume":"18","author":"Maurer","year":"2012","journal-title":"Bernoulli"},{"key":"2026032712265640800_ref135","volume-title":"Statistical Physics and Information Theory, volume 6 of Foundations and Trends in Communications and Information Theory","author":"Merhav","year":"2009"},{"issue":"3","key":"2026032712265640800_ref136","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1214\/aop\/1055425791","article-title":"Concentration inequalities using the entropy method","volume":"31","author":"Boucheron","year":"2003","journal-title":"Annals of Probability"},{"key":"2026032712265640800_ref137","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1214\/ECP.v11-1190","article-title":"Measure concentration for compound Poisson distributions","volume":"11","author":"Kontoyiannis","year":"2006","journal-title":"Electronic Communications in Probability"},{"key":"2026032712265640800_ref138","doi-asserted-by":"crossref","first-page":"586","DOI":"10.1214\/aos\/1176345462","article-title":"The jackknife estimate of variance","volume":"9","author":"Efron","year":"1981","journal-title":"Annals of Statistics"},{"key":"2026032712265640800_ref139","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1214\/aos\/1176349952","article-title":"An Efron\u2013Stein inequality for nonsymmetric statistics","volume":"14","author":"Steele","year":"1986","journal-title":"Annals of Statistics"},{"key":"2026032712265640800_ref140","volume-title":"Metric Structures for Riemannian and Non-Riemannian Spaces","author":"Gromov","year":"2001"},{"key":"2026032712265640800_ref141","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1006\/jfan.1996.0002","article-title":"A functional form of the isoperimetric inequality for the Gaussian measure","volume":"135","author":"Bobkov","year":"1996","journal-title":"Journal of Functional Analysis"},{"issue":"4","key":"2026032712265640800_ref142","doi-asserted-by":"crossref","first-page":"1381","DOI":"10.1007\/s10958-006-0049-2","article-title":"On the translocation of masses","volume":"133","author":"Kantorovich","year":"2006","journal-title":"Journal of Mathematical Sciences"},{"issue":"5","key":"2026032712265640800_ref143","doi-asserted-by":"crossref","first-page":"1836","DOI":"10.1109\/TIT.2005.846407","article-title":"A distribution dependent refinement of Pinsker\u2019s inequality","volume":"51","author":"Ordentlich","year":"2005","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref144","volume-title":"Inequalities for the L1 deviation of the empirical distribution","author":"Weissman","year":"2003"},{"key":"2026032712265640800_ref145","unstructured":"D.\n              Berend\n            , P.Harremo\u0451s, and A.Kontorovich. A reverse Pinsker inequality, 2012. URL: http:\/\/arxiv.org\/abs\/1206.6544."},{"key":"2026032712265640800_ref146","first-page":"1","author":"Sason","year":"2013"},{"issue":"2","key":"2026032712265640800_ref147","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1109\/TIT.2004.840861","article-title":"Entropy and the law of small numbers","volume":"51","author":"Kontoyiannis","year":"2005","journal-title":"IEEE Trans. on Information Theory"},{"issue":"3","key":"2026032712265640800_ref148","doi-asserted-by":"crossref","first-page":"768","DOI":"10.1214\/aop\/1176993227","article-title":"Sanov property, generalized I-projection and a conditional limit theorem","volume":"12","author":"Csisz\u00e1r","year":"1984","journal-title":"Annals of Probability"},{"key":"2026032712265640800_ref149","doi-asserted-by":"crossref","DOI":"10.1002\/9781118165904","volume-title":"A Weak Convergence Approach to the Theory of Large Deviations","author":"Dupuis","year":"1997"},{"issue":"3","key":"2026032712265640800_ref150","doi-asserted-by":"crossref","first-page":"587","DOI":"10.1007\/BF02249265","article-title":"Transportation cost for Gaussian and other product measures","volume":"6","author":"Talagrand","year":"1996","journal-title":"Geometry and Functional Analysis"},{"key":"2026032712265640800_ref151","volume-title":"Real Analysis and Probability","author":"Dudley","year":"2004"},{"key":"2026032712265640800_ref152","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1006\/jfan.1999.3557","article-title":"Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality","volume":"173","author":"Otto","year":"2000","journal-title":"Journal of Functional Analysis"},{"key":"2026032712265640800_ref153","volume-title":"On the HWI inequality. a work in progress","author":"Wu"},{"key":"2026032712265640800_ref154","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1007\/s002050100185","article-title":"Some applications of mass transport to Gaussian-type inequalities","volume":"161","author":"Cordero-Erausquin","year":"2002","journal-title":"Arch. Rational Mech. Anal."},{"key":"2026032712265640800_ref155","first-page":"177","volume-title":"S\u00e9minaire de Probabilit\u00e9s XIX, volume 1123 of Lecture Notes in Mathematics","author":"Bakry","year":"1985"},{"issue":"1","key":"2026032712265640800_ref156","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1214\/aop\/1019160125","article-title":"Concentration of measure inequalities for Markov chains and \u03c6-mixing processes","volume":"28","author":"Samson","year":"2000","journal-title":"Annals of Probability"},{"key":"2026032712265640800_ref157","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1007\/BF02249263","article-title":"A measure concentration inequality for contracting Markov chains","volume":"6","author":"Marton","year":"1996","journal-title":"Geometric and Functional Analysis"},{"issue":"3B","key":"2026032712265640800_ref158","doi-asserted-by":"crossref","first-page":"2526","DOI":"10.1214\/009117904000000702","article-title":"Measure concentration for Euclidean distance in the case of dependent random variables","volume":"32","author":"Marton","year":"2004","journal-title":"Annals of Probability"},{"issue":"1","key":"2026032712265640800_ref159","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1214\/09-AOP463","article-title":"Correction to \u2018Measure concentration for Euclidean distance in the case of dependent random variables\u2019","volume":"38","author":"Marton","year":"2010","journal-title":"Annals of Probability"},{"key":"2026032712265640800_ref160","unstructured":"K.\n              Marton\n            \n          . Bounding relative entropy by the relative entropy of local specifications in product spaces, 2009. URL: http:\/\/arxiv.org\/abs\/0907.4491."},{"key":"2026032712265640800_ref161","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.jfa.2012.10.001","article-title":"An inequality for relative entropy and logarithmic Sobolev inequalities in Euclidean spaces","volume":"264","author":"Marton","year":"2013","journal-title":"Journal of Functional Analysis"},{"key":"2026032712265640800_ref162","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511921889","volume-title":"Information Theory: Coding Theorems for Discrete Memoryless Systems","author":"Csisz\u00e1r","year":"2011","edition":"2nd"},{"issue":"2","key":"2026032712265640800_ref163","first-page":"174","article-title":"Probabilistic characteristics of graphs with large connectivity","volume":"10","author":"Margulis","year":"1974","journal-title":"Problems of Information Transmission"},{"key":"2026032712265640800_ref164","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139030687","volume-title":"Network Information Theory","author":"El Gamal","year":"2011"},{"issue":"1","key":"2026032712265640800_ref165","first-page":"11","article-title":"Maximal error capacity regions are smaller than average error capacity regions for multi-user channels","volume":"7","author":"Dueck","year":"1978","journal-title":"Problems of Control and Information Theory"},{"issue":"4","key":"2026032712265640800_ref166","first-page":"339","article-title":"The maximal-error and average-error capacity regions for the broadcast channels are identical: a direct proof","volume":"19","author":"Willems","year":"1990","journal-title":"Problems of Control and Information Theory"},{"issue":"6","key":"2026032712265640800_ref167","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1109\/TIT.1975.1055469","article-title":"Source coding with side information and a converse for degraded broadcast channels","volume":"21","author":"Ahlswede","year":"1975","journal-title":"IEEE Trans. on Information Theory"},{"issue":"3","key":"2026032712265640800_ref168","doi-asserted-by":"crossref","first-page":"836","DOI":"10.1109\/18.568695","article-title":"The empirical distribution of good codes","volume":"43","author":"Shamai","year":"1997","journal-title":"IEEE Trans. on Information Theory"},{"issue":"3","key":"2026032712265640800_ref169","doi-asserted-by":"crossref","first-page":"752","DOI":"10.1109\/18.256486","article-title":"Approximation theory of output statistics","volume":"39","author":"Han","year":"1993","journal-title":"IEEE Trans. on Information Theory"},{"key":"2026032712265640800_ref170","unstructured":"Y.\n              Polyanskiy\n             and S.Verd\u00fa. Empirical distribution of good channel codes with non-vanishing error probability. To appear in the IEEE Trans. on Information Theory. URL: http:\/\/arxiv.org\/abs\/1309.0141."},{"key":"2026032712265640800_ref171","first-page":"291","article-title":"An information theoretical identity and a problem involving capacity","volume":"2","author":"Tops\u00f8e","year":"1967","journal-title":"Studia Scientiarum Mathematicarum Hungarica"},{"key":"2026032712265640800_ref172","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/1385-7258(74)90000-6","article-title":"On the Shannon capacity of an arbitrary channel","volume":"36","author":"Kemperman","year":"1974","journal-title":"Indagationes Mathematicae"},{"key":"2026032712265640800_ref173","doi-asserted-by":"crossref","first-page":"10","DOI":"10.1007\/BF00531808","article-title":"Ged\u00e4chtnisfreie Kan\u00e4le f\u00fcr diskrete Zeit","volume":"6","author":"Augustin","year":"1966","journal-title":"Z. Wahrschein-lichkeitstheorie verw. Gebiete"},{"issue":"3","key":"2026032712265640800_ref174","first-page":"216","article-title":"An elementary proof of the strong converse theorem for the multiple-access channel","volume":"7","author":"Ahlswede","year":"1982","journal-title":"Journal of Combinatorics, Information and System Sciences"},{"issue":"12","key":"2026032712265640800_ref175","doi-asserted-by":"crossref","first-page":"3029","DOI":"10.1109\/TIT.2002.805064","article-title":"Variations on the Gallager bounds, connections and applications","volume":"48","author":"Shamai","year":"2001","journal-title":"IEEE Trans. on Information Theory"},{"issue":"4","key":"2026032712265640800_ref176","doi-asserted-by":"crossref","first-page":"1544","DOI":"10.1109\/18.923735","article-title":"Sphere-covering, measure concentration, and source coding","volume":"47","author":"Kontoyiannis","year":"2001","journal-title":"IEEE Trans. on Information Theory"},{"issue":"5","key":"2026032712265640800_ref177","doi-asserted-by":"crossref","first-page":"1850","DOI":"10.1109\/TIT.2008.920242","article-title":"State amplification","volume":"54","author":"Kim","year":"2008","journal-title":"IEEE Trans. on Information Theory"}],"container-title":["Foundations and Trends\u00ae in Communications and Information Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.emerald.com\/ftcit\/article-pdf\/10\/1-2\/1\/11153899\/0100000064en.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/www.emerald.com\/ftcit\/article-pdf\/10\/1-2\/1\/11153899\/0100000064en.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T16:27:29Z","timestamp":1774628849000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.emerald.com\/ftcit\/article\/10\/1-2\/1\/1332380\/Concentration-of-Measure-Inequalities-in"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,23]]},"references-count":177,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2013,10,23]]}},"URL":"https:\/\/doi.org\/10.1561\/0100000064","relation":{},"ISSN":["1567-2190","1567-2328"],"issn-type":[{"value":"1567-2190","type":"print"},{"value":"1567-2328","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,10,23]]}}}