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This page intentionally left blank FINITARY PROBABILISTIC METHODS IN ECONOPHYSICS Econophysics applies the methodology of physics to the study of economics. However, whilst physicists have a good understanding of statistical physics, they areoftenunfamiliarwithrecentadvancesinstatistics,includingBayesianandpre- dictivemethods.Equally,economistswithknowledgeofprobabilitiesdonothave a background in statistical physics and agent-based models. Proposing a unified viewforadynamicprobabilisticapproach,thisbookisusefulforadvancedunder- graduate and graduate students as well as researchers in physics, economics and finance. The book takes a finitary approach to the subject. It discusses the essentials of applied probability, and covers finite Markov chain theory and its applications to real systems. Each chapter ends with a summary, suggestions for further reading andexerciseswithsolutionsattheendofthebook. Ubaldo Garibaldi is First Researcher at the IMEM-CNR, Italy, where he researches the foundations of probability, statistics and statistical mechanics, and theapplicationoffiniteMarkovchainstocomplexsystems. Enrico Scalas isAssistant Professor of Physics at the University of Eastern Piedmont, Italy. His research interests are anomalous diffusion and its applica- tionstocomplexsystems,thefoundationsofstatisticalmechanicsandagent-based simulationsinphysics,financeandeconomics. FINITARY PROBABILISTIC METHODS IN ECONOPHYSICS UBALDO GARIBALDI IMEM-CNR,Italy ENRICO SCALAS UniversityofEasternPiedmont,Italy CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521515597 © U. Garibaldi and E. Scalas 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13 978-0-511-90205-5 eBook (NetLibrary) ISBN-13 978-0-521-51559-7 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Foreword page ix Acknowledgements xiii 1 Introductoryremarks 1 1.1 Earlyaccountsandthebirthofmathematicalprobability 1 1.2 Laplaceandtheclassicaldefinitionofprobability 3 1.3 Frequentism 8 1.4 Subjectivism,neo-Bayesianismandlogicism 10 1.5 Fromdefinitionstointerpretations 12 Furtherreading 13 References 13 2 Individualandstatisticaldescriptions 15 2.1 Jointα-descriptionsands-descriptions 15 2.2 Frequencyvectorandindividualdescriptions 18 2.3 Partitions 19 2.4 Partialandmarginaldescriptions 20 2.5 Exercises 24 2.6 Summary 24 Furtherreading 25 3 Probabilityandevents 26 3.1 Elementarydescriptionsandevents 26 3.2 Decompositionsofthesamplespace 29 3.3 Remarksondistributions 32 3.4 Probabilityassignments 35 3.5 Remarksonurnmodelsandpredictiveprobabilities 39 3.6 Appendix:outlineofelementaryprobabilitytheory 42 3.7 Exercises 54 3.8 Summary 55 Furtherreading 55 References 57 v vi Contents 4 Finiterandomvariablesandstochasticprocesses 58 4.1 Finiterandomvariables 58 4.2 Finitestochasticprocesses 73 4.3 Appendix1:afiniteversionofdeFinetti’stheorem 86 4.4 Appendix2:thebetadistribution 92 4.5 Exercises 93 4.6 Summary 93 Furtherreading 94 References 95 5 ThePólyaprocess 96 5.1 DefinitionofthePólyaprocess 96 5.2 ThefirstmomentsofthePólyadistribution 100 5.3 Labelmixingandmarginaldistributions 104 (cid:1) 5.4 Aconsequenceoftheconstraint n =n 116 i 5.5 ThecontinuumlimitsofthemultivariatePólyadistribution 116 5.6 Thefundamentalrepresentationtheoremforthe Pólyaprocess 123 5.7 Exercises 130 5.8 Summary 131 Furtherreading 132 6 TimeevolutionandfiniteMarkovchains 134 6.1 FromkinematicstoMarkovianprobabilisticdynamics 134 6.2 TheEhrenfesturnmodel 139 6.3 FiniteMarkovchains 141 6.4 Convergencetoalimitingdistribution 146 6.5 Theinvariantdistribution 152 6.6 Reversibility 160 6.7 Exercises 166 6.8 Summary 169 Furtherreading 169 References 171 7 TheEhrenfest–Brillouinmodel 172 7.1 MergingEhrenfest-likedestructionsand Brillouin-likecreations 172 7.2 Unarymoves 174 7.3 Fromfleastoants 177 7.4 Morecomplicatedmoves 180 7.5 Pólyadistributionstructures 181 7.6 Anapplicationtostockpricedynamics 188 7.7 Exogenousconstraintsandthemostprobable occupationvector 193 Contents vii 7.8 Exercises 201 7.9 Summary 201 Furtherreading 202 8 Applicationstostylizedmodelsineconomics 204 8.1 Amodelforrandomcoinexchange 204 8.2 Thetaxation–redistributionmodel 212 8.3 TheAoki–Yoshikawamodelforsectoralproductivity 217 8.4 Generalremarksonstatisticalequilibriumineconomics 223 8.5 Exercises 225 8.6 Summary 225 Furtherreading 226 References 228 9 FinitarycharacterizationoftheEwenssamplingformula 229 9.1 Infinitenumberofcategories 229 9.2 FinitaryderivationoftheEwenssamplingformula 232 9.3 Clusternumberdistribution 238 9.4 Ewens’momentsandsite-labelmarginals 240 9.5 Alternativederivationoftheexpectednumberofclusters 243 9.6 Samplingandaccommodation 244 9.7 Markovchainsforclusterandsitedynamics 247 9.8 Marginalclusterdynamicsandsitedynamics 250 9.9 Thetwo-parameterEwensprocess 256 9.10 Summary 259 Furtherreading 260 10 TheZipf–Simon–Yuleprocess 262 10.1 TheEwenssamplingformulaandfirmsizes 262 10.2 Hoppe’svs.Zipf’surn 263 10.3 ExpectedclustersizedynamicsandtheYuledistribution 265 10.4 BirthanddeathSimon–Zipf’sprocess 269 10.5 MarginaldescriptionoftheSimon–Zipfprocess 271 10.6 Aformalreversiblebirth-and-deathmarginalchain 274 10.7 MonteCarlosimulations 276 10.8 Continuouslimitanddiffusions 279 10.9 Appendix1:invariantmeasureforhomogeneousdiffusions 286 10.10 Summary 287 Furtherreading 288 Appendix 289 Solutionstoexercises 289 Authorindex 323 Subjectindex 325