An Introduction to Heavy-Tailed and Subexponential Distributions

166 Pages·2013·1.416 MB·English

by  Sergey Foss, Dmitry Korshunov, Stan Zachary (auth.)

Springer Series in Operations Research
and Financial Engineering
Sergey Foss
Dmitry Korshunov
Stan Zachary
An Introduction
to Heavy-Tailed
and Subexponential
Distributions
Second Edition
Springer Series in Operations Research
and Financial Engineering
SeriesEditors:
ThomasV.Mikosch
SidneyI.Resnick
StephenM.Robinson
Forfurthervolumes:
http://www.springer.com/series/3182

Sergey Foss • Dmitry Korshunov • Stan Zachary
An Introduction to
Heavy-Tailed and
Subexponential Distributions
Second Edition
123
SergeyFoss DmitryKorshunov
DepartmentofActuarialMathematics SobolevInstituteofMathematics
Heriot-WattUniversity Novosibirsk,Russia
Riccarton,Edinburgh,UK
StanZachary
DepartmentofActuarialMathematics
Heriot-WattUniversity
Riccarton,Edinburgh,UK
ISSN1431-8598
ISBN978-1-4614-7100-4 ISBN978-1-4614-7101-1(eBook)
DOI10.1007/978-1-4614-7101-1
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013937950
MathematicsSubjectClassification(2010):60E99,62E20,60F10,60G50
©SpringerScience+BusinessMediaNewYork2013
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Preface to the First Edition
This text studies heavy-tailed distributions in probability theory, and especially
convolutionsofsuchdistributions.Themaingoalistoprovideacompleteandcom-
prehensiveintroductiontothetheoryoflong-tailedandsubexponentialdistributions
whichincludesmanynovelelementsand,inparticular,isbasedontheregularuse
oftheprincipleofasinglebigjump.Muchofthematerialappearsforthefirsttime
intextform,including:
– The establishment of new relations between known classes of subexponential
distributionsandtheintroductionofimportantnewclasses
– The development of some important new concepts, including those of
h-insensitivityandlocalsubexponentiality
– The presentation of new and direct probabilistic proofs of known asymptotic
results
A number of recent textbooks and monographs contain some elements of the
presenttheory,notablythosebyS.Asmussen[1,2],P.Embrechts,C.Klu¨ppelberg,
and T. Mikosch [24], T. Rolski, H. Schmidli, V. Schmidt, and J. Teugels [47],
and A. Borovkovand K. Borovkov[11]. Further,the monographby N. Bingham,
C. Goldie, and J. Teugels [9] comprehensively develops the theory of regularly
varying functions and distributions; the latter form an important subclass of the
subexponentialdistributions.We have beeninfluencedby these booksand by fur-
thercontactswiththeirauthors.
Chapters2and3ofthepresentmonographdealcomprehensivelywiththeprop-
ertiesofheavy-tailed,long-tailedandsubexponentialdistributions,andgiveappli-
cationstorandomsums.Chapter4developsconceptsoflocalsubexponentialityand
givesfurtherapplications.Finally,Chap.5studiesthedistributionofthemaximum
of a randomwalk with negativedriftand heavy-tailedincrements;notablyit con-
tainsnewandshortprobabilisticproofsforthetailasymptoticsofthisdistribution
forbothfiniteandinfinitetimehorizons.Thestudyofheavy-taileddistributionsin
more general probabilitymodels—for example, Markov-modulatedmodels, those
with dependencies, and continuous-time models—is postponed until such future
date as the authorsmay again find some spare time. Nevertheless, the same basic
principlesapplythereasaredevelopedinthepresenttext.
v
vi PrefacetotheFirstEdition
Theauthorsgratefullyacknowledgeafruitfulcollaborationonheavy-tailsissues
with their co-authors: Søren Asmussen, Franc¸ois Baccelli, Aleksandr Borovkov,
Onno Boxma, Denis Denisov, Takis Konstantopoulos, Marc Lelarge, Andrew
Richards,andVolkerSchmidt.
Wearethankfultomanycolleagues,inadditiontomentionedabove,forhelpful
discussionsandcontributions,notablytoVsevolodShneerandBertZwart.Wethank
SergeiFedotovforpointingoutlinkstoanalogousproblemsinStatisticalPhysics.
We are also verygratefulboth to ThomasMikosch andto the staff of Springer
fortheirsuggestionsandassistanceinpublishingthistext.
This book was mostly written while the authors worked, together and indi-
vidually, in Edinburgh and Novosibirsk; we thank our home institutions, Heriot-
Watt University and the Sobolev Institute of Mathematics. A first version of this
manuscript was finished while the authors stayed at the Mathematisches
Forschungsinstitut Oberwolfach, under the Research in Pairs programme from
March 23 to April 5, 2008; we thank the Institute for its great hospitality and
support.Thefinalversionwaspreparedin Cambridgeduringourstay atthe Isaac
NewtonInstituteforMathematicalSciencesundertheframeworkoftheprogramme
StochasticProcessesinCommunicationSciences,January–June,2010.
Alistoferrataandnotesonfurtherdevelopmentstothismanuscriptwillbemain-
tainedathttps://sites.google.com/site/ithtsd/https://sites.google.com/site/ithtsd/.
Edinburgh SergeyFoss
Novosibirsk DmitryKorshunov
OberwolfachandCambridge StanZachary
Preface
ThisisanextendedandcorrectedversionoftheFirstEdition.Themajorchangesare:
– Chapters 2 through 5 are now appended by lists of problems and exercises.
Wealsoprovideanswersandanumberofsolutions.
– Chapter5includesthreenewsectionsonapplications,toqueueingtheory,torisk,
andtobranchingprocesses,anda newsection describingtimeto exceeda high
levelbyarandomwalkanditslocationaroundthattime.
– Sections5.1,5.2and5.9areextended.
SergeyFoss
August2012 DmitryKorshunov
vii

Contents
1 Introduction................................................... 1
2 Heavy-TailedandLong-TailedDistributions ...................... 7
2.1 Heavy-TailedDistributions................................... 7
2.2 CharacterisationofHeavy-TailedDistributions
inTermsofGeneralisedMoments............................. 11
2.3 LowerLimitforTailsofConvolutions ......................... 14
2.4 Long-TailedFunctionsandTheirProperties .................... 17
2.5 Long-TailedDistributions.................................... 21
2.6 Long-TailedDistributionsandIntegratedTails .................. 22
2.7 ConvolutionsofLong-TailedDistributions ..................... 24
2.8 h-InsensitiveDistributions ................................... 31
2.9 Comments ................................................ 38
2.10 Problems ................................................. 39
3 SubexponentialDistributions.................................... 43
3.1 SubexponentialDistributionsonthePositiveHalf-Line........... 43
3.2 SubexponentialDistributionsontheWholeRealLine ............ 45
3.3 SubexponentialityandWeakTail-Equivalence .................. 49
3.4 TheClassS∗ ofStrongSubexponentialDistributions............. 53
3.5 SufficientConditionsforSubexponentiality..................... 57
3.6 ConditionsforSubexponentialityinTerms
ofTruncatedExponentialMoments ........................... 59
3.7 SIsaProperSubsetofL .................................... 62
3.8 DoesF ∈SImplyThatF ∈S?............................... 63
I
3.9 ClosurePropertiesoftheClassofSubexponentialDistributions.... 65
3.10 Kesten’sBound ............................................ 67
3.11 SubexponentialityandRandomlyStoppedSums ................ 69
3.12 Comments ................................................ 71
3.13 Problems ................................................. 72
ix