Chance and Stability. Stable Distributions and their Applications

594 Pages·1999·8.151 MB·English

by  Vladimir M. Zolotarev, Vladimir V. Uchaikin

CHANCE and STABILITY
Stable Distributions and their Applications
Vladimir V. Uchaikin Vladimir M. Zolotarev
c VSP1999
(cid:13)
Fluctuat nec mergitur
Contents
Foreword vii
Introduction xi
I Theory 1
1 Probability 3
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Probabilityspace 3
. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Randomvariables 4
l . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 FunctionsX( ) 5
. . . . . . . . . . . . . . . . . . . .
1.4 Randomvectorsandprocesses 7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5 Independence 10
. . . . . . . . . . . . . . . . . . . . . . . . . .
1.6 Meanandvariance 12
. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.7 Bernoullitheorem 14
. . . . . . . . . . . . . . . . . . . .
1.8 TheMoivre–Laplacetheorem 20
. . . . . . . . . . . . . . . . . . . . . . .
1.9 Thelawoflargenumbers 22
. . . . . . . . . . . . . . . . . . . . .
1.10 Stronglawoflargenumbers 24
. . . . . . . . . . . . . . . . . . . . . .
1.11 Ergodicityandstationarity 28
. . . . . . . . . . . . . . . . . . . . . . .
1.12 Thecentrallimittheorem 31
2 Elementaryintroductiontothetheoryofstablelaws 35
. . . . . . . . . . . . . . . . . . . .
2.1 Convolutionsofdistributions 35
. . . . . . . . .
2.2 TheGaussdistributionandthestabilityproperty 39
2.3 TheCauchyandLe´vydistributions . . . . . . . . . . . . . . . . . 44
. . . . . . . . . .
2.4 Summationofstrictlystablerandomvariables 50
. . . . . . . . . . . . . .
2.5 Thestablelawsaslimitingdistributions 55
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Summary 64
3 Characteristicfunctions 69
. . . . . . . . . . . . . . . . . . . . . . . .
3.1 Characteristicfunctions 69
. .
3.2 Thecharacteristicfunctionsofsymmetricstabledistributions 72
a . . . . . . . . . . . . . . . .
3.3 Skewstabledistributionswith < 1 77
i
. . . . . . . .
3.4 Thegeneralformofstablecharacteristicfunctions 82
. . . . . . . . . . . . . . .
3.5 Stablelawsasinfinitelydivisiblelaws 86
. . . . . . . . . .
3.6 Variousformsofstablecharacteristicfunctions 93
. . . . . . . . . . . .
3.7 Somepropertiesofstablerandomvariables 98
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Conclusion 102
4 Probabilitydensities 103
. . . . . . . . . . . . . . . . . . . . . . .
4.1 Symmetricdistributions 103
. . . . . . . . . .
4.2 Convergentseriesforasymmetricdistributions 105
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Longtails 110
. . . . . . . . . . . . .
4.4 Integralrepresentationofstabledensities 115
. . . . . .
4.5 Integralrepresentationofstabledistributionfunctions 119
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Dualitylaw 121
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Shorttails 123
a . . . . . . . .
4.8 Stabledistributionswith closetoextremevalues 127
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.9 Summary 130
5 Integraltransformations 137
. . . . . . . . . . . . . . . . . . . . . . . .
5.1 Laplacetransformation 137
. . . . . . . . . . . . . .
5.2 InversionoftheLaplacetransformation 140
. . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Tauberiantheorems 143
. . . . . . . . . . . . . . . . . . . .
5.4 One-sidedstabledistributions 145
. . . . . . . . .
5.5 Laplacetransformationoftwo-sideddistributions 150
. . . . . . . . . . . . . . . . . . . . . .
5.6 TheMellintransformation 152
. . . . . . . . . . . . . . . . . .
5.7 Thecharacteristictransformation 154
. . . . . . . . . . . . . . . . . . . . . . .
5.8 Thelogarithmicmoments 156
. . . . . . . . . . . . . . . .
5.9 Multiplicationanddivisiontheorems 158
6 Specialfunctionsandequations 167
. . . . . . . . . . . . . . . . . . . . .
6.1 Integrodifferentialequations 167
. . . . . . . . . . . . . . . . . . . . . . . . .
6.2 TheLaplaceequation 169
. . . . . . . . . . . . . .
6.3 Fractionalintegrodifferentialequations 171
. . . . . . . . . . . . . . . .
6.4 Splittingofthedifferentialequations 174
. . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Somespecialcases 176
. . . . . . . . . . . . . . . . . . . . . . .
6.6 TheWhittakerfunctions 178
. . . . . . . .
6.7 Generalizedincompletehypergeometricalfunction 179
. . . . . . . . . . . . . . . . . . . .
6.8 TheMeijerandFoxfunctions 181
. . . . . . . . . . .
6.9 Stabledensitiesasaclassofspecialfunctions 185
. . . . . . . . . . . . . . . . . . . . . . . . .
6.10 Transstablefunctions 188
. . . . . . . . . . . . . . . . . . . . . . . . . .
6.11 Concludingremarks 190
ii
7 Multivariatestablelaws 193
. . . . . . . . . . . . . . . . . . . .
7.1 Bivariatestabledistributions 193
. . . . . . . . . . . . . . . . . . . .
7.2 Trivariatestabledistributions 200
. . . . . . . . . . . . . . . . . .
7.3 Multivariatestabledistributions 203
. . . . . . . . .
7.4 Sphericallysymmetricmultivariatedistributions 207
. . . . . . . . . . . . .
7.5 Sphericallysymmetricstabledistributions 210
8 Simulation 213
. . . . . . . . . . . . . . . . . . . . .
8.1 Theinversefunctionmethod 213
. . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Thegeneralformula 216
8.3 Approximate algorithm for one-dimensional symmetric stable
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
variables 218
8.4 Simulation of three-dimensional spherically symmetric stable
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vectors 221
9 Estimation 229
. . . . . . . . . . . . . . . . . . . . . . .
9.1 Samplefractiletechnique 229
. . . . . . . . . . . . . . . . . .
9.2 Methodofcharacteristicfunctions 231
n q t . .
9.3 Methodofcharacteristictransforms: estimatorsof , and 232
a . . . . . . . . . . . . . . . . . . . . . . .
9.4 Invariantestimationof 240
g . . . . . . . . . . . . . . . . . . . . . .
9.5 Estimatorsofparameter 241
. . . . . . . . . . . . . . . . . . .
9.6 Maximumlikelihoodestimators 245
a . . . . . . . . . . . . . . . . .
9.7 Fisher’sinformationfor closeto2 249
. . . . . . . . . . . . . . . . . . . . . . . . . .
9.8 Concludingremarks 251
II Applications 253
10 Someprobabilisticmodels 255
. . . . . . . . . . . . . . . . . . . . . . . . .
10.1 Generatingfunctions 255
. . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Stablelawsingames 257
. . . . . . . . . . . . . . . . . . . . .
10.3 Randomwalksanddiffusion 261
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Stableprocesses 265
. . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Branchingprocesses 274
. . . . . . . . . . . . . . . .
10.6 Pointsources: two-dimensionalcase 281
. . . . . . . . . . . . . . . .
10.7 Pointsources: multidimensionalcase 286
. . . . . . . . .
10.8 Aclassofsourcesgeneratingstabledistributions 290
11 Correlatedsystemsandfractals 297
. . . . .
11.1 Randompointdistributionsandgeneratingfunctionals 297
. . . . . . . . . . . . . . . . . . . . . .
11.2 Markovpointdistributions 301
. . . . . . . . . . . . . . .
11.3 Averagedensityofrandomdistribution 303
. . . . . . . . . . . . . . . . . . . . . . . . .
11.4 Correlationfunctions 306
. . . . .
11.5 Inversepowertypecorrelationsandstabledistributions 310
. . . . . . . . . . . . . . . . . . .
11.6 Mandelbrot’sstochasticfractals 315
iii
. . . . . . . . . . . . . . . . . . . . . . . . . . .
11.7 Numericalresults 319
. . . . . . . . . . . .
11.8 Fractalsetswithaturnovertohomogeneity 322
12 Anomalousdiffusionandchaos 331
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.1 Introduction 331
. . . . . . . . . . . . . . . .
12.2 Twoexamplesofanomalousdiffusion 333
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Superdiffusion 336
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4 Subdiffusion 343
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.5 CTRWequations 349
. . . . . . . . . . . . . . . . . . . . . . . . . . .
12.6 Somespecialcases 352
. . . . . . . .
12.7 AsymptoticsolutionoftheMontroll–Weissproblem 356
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.8 Two-statemodel 358
. . . . . . . . . . . . . . . . . . . . . . . . . .
12.9 Stablelawsinchaos 361
13 Physics 365
. . . . . . . . . . . . . . . . . . . . . . .
13.1 Lorentzdispersionprofile 365
. .
13.2 Starkeffectinanelectricalfieldofrandomlydistributedions 367
. . . . . . . . . . . . . . . . . . . . . . .
13.3 Dipolesandquadrupoles 372
. . . . . . . . . . . . . . . . . . . . . . . . . .
13.4 Landaudistribution 373
. . . . . . . . . . . . . . .
13.5 Multiplescatteringofchargedparticles 377
. . . . . . . . . . . . . . . . . . . . . . . . . . .
13.6 Fractalturbulence 381
. . . . . . . . . . . . . . . . . . . .
13.7 Stressesincrystallinelattices 382
. . . . . . . . .
13.8 Scale-invariantpatternsinacicularmartensites 383
. . . . . . . . . . . . . . . . . . . .
13.9 Relaxationinglassymaterials 384
. . . . . . . . . . . . . . . . . . . . . . . .
13.10Quantumdecaytheory 386
. . . . . . . . . . . . . . .
13.11Localizedvibrationalstates(fractons) 389
. . . . . . . . . . . . . . .
13.12Anomaloustransit-timeinsomesolids 390
. . . . . . . . . . . . . . . . . . . . . . . . . .
13.13Latticepercolation 392
. . . . . . . . . . . . . . . . . . .
13.14Wavesinmediumwithmemory 394
. . . . . . . . . . . . . . . . . . . . . . . . .
13.15Themesoscopiceffect 397
. . . . . . . . . . . . . . . . . . . . . . .
13.16Multiparticleproduction 398
. . . . . . . . . . . . . . . . . . . . . . . . .
13.17Tsallis’distributions 400
. . . . . . . . . .
13.18Stabledistributionsandrenormalizationgroup 402
14 Radiophysics 405
. . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1 Transmissionline 405
. . . . . . . . . . . . . . . . . . .
14.2 Distortionofinformationphase 407
. . . . . . . . . . . . .
14.3 Signalandnoiseinamultichannelsystem 410
. . . . . . . . . . . . . . . .
14.4 Wavescatteringinturbulentmedium 413
. . . . . . . . . . . . . . . . . . . . . . . . .
14.5 Chaoticphasescreen 416
iv
15 Astrophysicsandcosmology 419
. . . . . . . . . . . . . . . . . . . . . . . . .
15.1 Lightofadistantstar 419
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.2 Cosmicrays 420
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.3 Stellardynamics 424
. . . . . . . . . . . . . . . . . .
15.4 Cosmologicalmonopoleanddipole 428
. . . . . . . . . . . . . . . . . . .
15.5 TheUniverseasarippledwater 429
. . . . . . . . . . . . . . . . . . . .
15.6 Thepowerspectrumanalysis 431
. . . . . . . . . .
15.7 Cell-countdistributionforthefractalUniverse 434
. . . . . . . . . . . .
15.8 GlobalmassdensityforthefractalUniverse 435
16 Stochasticalgorithms 439
. . . . . . . . . . .
16.1 Monte-Carloestimatorswithinfinitevariance 439
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Fluxatapoint 441
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.3 Examples 443
16.4 Estimationofalinearfunctionalofasolutionofintegralequation445
. . . . . . . . . . . . . . . . . . . . . . . . . . .
16.5 Randommatrices 453
. . . . . . . . . . . . . . . . . . .
16.6 Randomsymmetricpolynomials 454
17 Financialapplications 463
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.1 Introduction 463
. . . . . . . . . . . . . . . . . . . . . . .
17.2 Moreonstableprocesses 464
. . . . . . . . . . . . . . . . . . . .
17.3 Multivariatestableprocesses 466
. . . . . . . . . . . . . . . . . . . . . . . . .
17.4 Stableportfoliotheory 469
. . . . . . . . . . . . . . . . . . . . . . .
17.5 Log-stableoptionpricing 474
. . . . . . . . . . . . . . .
17.6 Lowprobabilityandshort-livedoptions 481
. . . . . . . . . . . .
17.7 Parameterestimationandempiricalissues 482
18 Miscellany 489
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1 Biology 489
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.2 Genetics 493
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.3 Physiology 496
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.4 Ecology 498
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.5 Geology 499
Appendix 501
A.1 One-dimensionaldensitiesqA(x;a ,b ) . . . . . . . . . . . . . . . . 503
A.2 One-sideddistributionfunctionsGB(x;a ,1)multipliedby104 . . 510
A.3 One-sideddistributionsrepresentedbyfunction
F(y;a ) =104GB(y−1/a ;a ,1)(F(y;0) 104e−y) . . . . . . . . . . . 511
A.4 Thefunctiona 1/a q(a 1/a x;a ),where≡q(x;a )istheone-dimensional
. . . . . . . . . . . . . . . . . . . . . . .
symmetricstabledensity 513
r a
A.5 Radial functions (r; ) of two-dimensional axially symmetric
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
densities 514
v
r a
A.6 Radial functions (r; ) of three-dimensional spherically sym-
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
metricdensities 515
A.7 Strictlystabledensitiesexpressedviaelementaryfunctions,spe-
. . . . . . . . . . . . . . . . . . . .
cialfunctionsandquadratures 516
. . . . . . . . . . . . . .
A.8 Fractionalintegro-differentialoperators 518
A.9 Approximationof inverse distributionfunctionr(x) = F−1(x) for
simulation of three-dimensional random vectors with density
a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
q3(r; ) 522
. . . . . . . . . . . . . . . . . . . . . . . . .
A.10Somestatisticalterms 524
. . . . . . . .
A.11Someauxiliaryformulaeforstatisticalestimators 525
. . . . . . . . . . . . . . . . . . . . . . . .
A.12Functionalderivatives 526
Conclusion 531
Bibliography 533
Index 567
vi
Foreword
This book is the third in the series of monographs “Modern Probability and
Statistics”followingthebooks
•
V.M.Zolotarev,ModernTheoryofSummationofRandomVariables;
•
V.V. Senatov, NormalApproximation: NewResults,MethodsandProb-
lems.
Thescopeoftheseriescanbeseenfromboththetitleoftheseriesandthe
titlesofthepublishedandforthcomingbooks:
•
Yu.S. Khokhlov, Generalizationsof StableDistributions: Structureand
LimitTheorems;
•
V. E. Bening, AsymptoticTheory of Testing StatisticalHypotheses: Effi-
cientStatistics,Optimality,Deficiency;
•
N.G.Ushakov,SelectedTopicsinCharacteristicFunctions.
Amongtheproposalsunderdiscussionarethefollowingbooks:
•
G.L.ShevlyakovandN.O.Vilchevskii,RobustEstimation: Criteriaand
Methods;
•
V. E. Bening, V. Yu. Korolev, and S. Ya. Shorgin, Compound Doubly
Stochastic Poisson Processes and Their Applications in Insurance and
Finance;
•
E.M.Kudlaev,DecomposableStatistics;
•
G. P. Chistyakov, Analytical Methods in the Problem of Stability of De-
compositionsofRandomVariables;
•
A.N.Chuprunov,RandomProcessesObservedatRandomTimes;
•
D.H.Mushtari,ProbabilitiesandTopologiesonLinearSpaces;
•
V.G.Ushakov,PriorityQueueingSystems;
vii
•
E. V. Morozov, GeneralQueueingNetworks: theMethodof Regenerative
Decomposition;
•
V. Yu. Korolev and V. M. Kruglov, Random Sequences with Random
Indices;
•
Yu. V. Prokhorov and A. P. Ushakova, Reconstruction of Distribution
Types;
•
L. Szeidl andV. M. Zolotarev, Limit Theoremsfor Random Polynomials
andRelatedTopics;
•
A.V. BulinskiiandM.A. Vronskii,Limit Theoremsfor AssociatedRan-
domVariables;
•
E.V.Bulinskaya,StochasticInventorySystems: FoundationsandRecent
Advances;
aswellasmanyothers.
Toprovidehigh-qualifiedinternationalexaminationoftheproposedbooks,
we invited well-known specialists to join the Editorial Board. All of them
kindlyagreed,sonowtheEditorialBoardoftheseriesisasfollows:
A.Balkema(UniversityofAmsterdam,theNetherlands)
W.Hazod(UniversityofDortmund,Germany)
V.Kalashnikov(MoscowInstituteforSystemsResearch,Russia)
V.Korolev(MoscowStateUniversity,Russia)—Editor-in-Chief
V.Kruglov(MoscowStateUniversity,Russia)
J.D.Mason(UniversityofUtah,SaltLakeCity,USA)
E.Omey(EHSAL,Brussels,Belgium)
K.Sato(NagoyaUniversity,Japan)
M.Yamazato(UniversityofRyukyu,Japan)
V.Zolotarev(Steklov MathematicalInstitute,Moscow,Russia)—Editor-
in-Chief
Wehopethatthebooksofthisserieswillbeinterestingandusefultoboth
specialistsinprobabilitytheory,mathematicalstatisticsandthoseprofession-
als who apply the methods and results of these sciences to solving practical
problems.
In our opinion, the present book to a great extent meets these require-
ments. An outbreak of interest to stable distributions is due to both their
analyticalpropertiesandimportantrole they playin variousapplicationsin-
cluding so different fields as, say, physics and finance. This book, written by
mathematician V. Zolotarev and physicist V. Uchaikin, can be regarded as a
comprehensiveintroductiontothetheory ofstabledistributionsandtheirap-
plications. It contains a modern outlook of the mathematical aspects of this
viii