Table Of ContentThe Fundamentals of Heavy Tails
Heavytails–extremeeventsorvaluesmorecommonthanexpected–emergeevery-
where:theeconomy,naturalevents,andsocialandinformationnetworksarejusta
fewexamples.Yetafterdecadesofprogress,theyarestilltreatedasmysterious,sur-
prising,andevencontroversial,primarilybecausethenecessarymathematicalmodels
andstatisticalmethodsarenotwidelyknown.
Thisbook,forthefirsttime,providesarigorousintroductiontoheavy-taileddistri-
butionsaccessibletoanyonewhoknowselementaryprobability.Ittacklesandtames
the zoo of terminology for models and properties, demystifying topics such as the
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genceofheavy-taileddistributionsfromawidevarietyofgeneralprocesses,building
intuition. And it reveals the controversy surrounding heavy tails to be the result of
flawedstatistics,thenequipsreaderstoidentifyandestimatewithconfidence.Over
100exercisescompletethisengagingpackage.
(cid:3459)(cid:3450)(cid:3474)(cid:3450)(cid:3460)(cid:3467)(cid:3458)(cid:3468)(cid:3457)(cid:3463)(cid:3450)(cid:3463) (cid:3463)(cid:3450)(cid:3458)(cid:3467)isAssociateProfessorinElectricalEngineeringatIITBom-
bay. His research focuses on modeling, performance evaluation, and design issues
in online learning environments, communication networks, queueing systems, and
smartpowergrids.HeistherecipientofbestpaperawardsatIFIPPerformance(2010
and2020)andACMe-Energy(2020).
(cid:3450)(cid:3453)(cid:3450)(cid:3462) (cid:3472)(cid:3458)(cid:3454)(cid:3467)(cid:3462)(cid:3450)(cid:3463)isProfessorofComputingandMathematicalSciencesattheCal-
ifornia Institute of Technology (Caltech). His research develops tools in machine
learning,optimization,control,andeconomicswiththegoalofmakingthenetworked
systemsthatgovernourworldsustainableandresilient.Heisbestknownforhiswork
spearheadingthedesignofalgorithmsforsustainabledatacentersandheistherecip-
ientofnumerousawardsincludingtheACMSigmetricsRisingStaraward,theACM
Sigmetrics Test of Time award, the IEEE Communication Society William Bennet
Prize,andmultipleteachingandbestpaperawards.
(cid:3451)(cid:3454)(cid:3467)(cid:3469) (cid:3475)(cid:3472)(cid:3450)(cid:3467)(cid:3469) is group leader at CWI Amsterdam and Professor of Mathematics
at Eindhoven University of Technology. He has expertise in stochastic operations
research,queueingtheory,andlargedeviations,andinthecontextofheavytails,he
has focused on sample path properties, designing Monte Carlo methods and appli-
cations to computer-communication and energy networks. He was area editor of
OperationsResearch,theflagshipjournalofhisprofession,from2009to2017,and
wastherecipientoftheINFORMSAppliedProbabilitySocietyErlangprize,awarded
everytwoyearstoanoutstandingyoungappliedprobabilist.
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The Fundamentals of Heavy Tails
Properties, Emergence, and Estimation
Jayakrishnan Nair
IndianInstituteofTechnology,Bombay
Adam Wierman
CaliforniaInstituteofTechnology
Bert Zwart
CWIAmsterdamandEindhovenUniversityofTechnology
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DOI:10.1017/9781009053730
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Contents
Preface pageix
Acknowledgments xiii
1 Introduction 1
1.1 DefiningHeavy-TailedDistributions 5
1.2 ExamplesofHeavy-TailedDistributions 9
1.3 What’sNext 24
1.4 Exercises 24
PartI Properties 27
2 ScaleInvariance,PowerLaws,andRegularVariation 29
2.1 ScaleInvarianceandPowerLaws 30
2.2 ApproximateScaleInvarianceandRegularVariation 32
2.3 AnalyticPropertiesofRegularlyVaryingFunctions 36
2.4 AnExample:ClosurePropertiesofRegularlyVaryingDistributions 48
2.5 AnExample:BranchingProcesses 50
2.6 AdditionalNotes 53
2.7 Exercises 54
3 Catastrophes,Conspiracies,andSubexponentialDistributions 56
3.1 ConspiraciesandCatastrophes 58
3.2 SubexponentialDistributions 62
3.3 AnExample:Randomsums 67
3.4 AnExample:ConspiraciesandCatastrophesinRandomWalks 72
3.5 AdditionalNotes 80
3.6 Exercises 81
4 ResidualLives,HazardRates,andLongTails 85
4.1 ResidualLivesandHazardRates 86
4.2 HeavyTailsandResidualLives 90
4.3 Long-TailedDistributions 93
4.4 AnExample:RandomExtrema 97
4.5 AdditionalNotes 100
4.6 Exercises 101
v
vi Contents
PartII Emergence 105
5 AdditiveProcesses 107
5.1 TheCentralLimitTheorem 108
5.2 GeneralizingtheCentralLimitTheorem 112
5.3 UnderstandingStableDistributions 114
5.4 TheGeneralizedCentralLimitTheorem 118
5.5 AVariation:TheEmergenceofHeavyTailsinRandomWalks 120
5.6 AdditionalNotes 124
5.7 Exercises 125
6 MultiplicativeProcesses 127
6.1 TheMultiplicativeCentralLimitTheorem 128
6.2 VariationsonMultiplicativeProcesses 131
6.3 AnExample:PreferentialAttachmentandYuleProcesses 138
6.4 AdditionalNotes 144
6.5 Exercises 145
7 ExtremalProcesses 148
7.1 ALimitTheoremforMaxima 150
7.2 UnderstandingMax-StableDistributions 154
7.3 TheExtremalCentralLimitTheorem 156
7.4 AnExample:ExtremesofRandomWalks 161
7.5 AVariation:TheTimebetweenRecordBreakingEvents 168
7.6 AdditionalNotes 170
7.7 Exercises 171
PartIII Estimation 175
8 EstimatingPower-LawDistributions:ListentotheBody 177
8.1 ParametricEstimationofPower-LawsUsingLinearRegression 179
8.2 MaximumLikelihoodEstimationforPower-LawDistributions 185
8.3 PropertiesoftheMaximumLikelihoodEstimator 187
8.4 VisualizingtheMLEviaRegression 189
8.5 ARecipeforParametricEstimationofPower-LawDistributions 192
8.6 AdditionalNotes 194
8.7 Exercises 195
9 EstimatingPower-LawTails:LettheTailDotheTalking 197
9.1 TheFailureofParametricEstimation 199
9.2 TheHillEstimator 203
9.3 PropertiesoftheHillEstimator 205
9.4 TheHillPlot 209
9.5 BeyondHillandRegularVariation 215
9.6 WhereDoestheTailBegin? 226
9.7 GuidelinesforEstimatingHeavy-TailedPhenomena 233
Contents vii
9.8 AdditionalNotes 235
9.9 Exercises 236
CommonlyUsedNotation 238
References 240
Index 249
