Bingham N.H., Goldie C.M., Teugels J.L. — Regular variation :: Электронная библиотека попечительского совета мехмата МГУ

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Bingham N.H., Goldie C.M., Teugels J.L. — Regular variation

Bingham N.H., Goldie C.M., Teugels J.L. — Regular variation

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Название: Regular variation

Авторы: Bingham N.H., Goldie C.M., Teugels J.L.

Аннотация:

Both the theory and applications of regular variation are given comprehensive coverage in this volume. In many limit theorems, regular variation is intrinsic to the result and exactly characterizes the limit behavior. The book emphasizes such characterizations, and gives a comprehensive treatment of those applications where regular variation plays an essential (rather than merely convenient) role. The authors rigorously develop the basic ideas of Karamata theory and de Haan theory including many new results and "second-order" theorems. They go on to discuss the role of regular variation in Abelian, Tauberian, and Mercerian theorems. These results are then applied in analytic number theory, complex analysis, and probability, with the aim of setting the theory in context. A widely scattered literature is thus brought together in a unified approach. With several appendices and a comprehensive list of references, analysts, number theorists, probabilitists, research workers, and graduate students will find this an invaluable and complete account of regular variation.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 494

Добавлена в каталог: 06.12.2009

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Предметный указатель

$G_{\delta}$ -set      62
$\emph{l}$ -index      see “g-index”
a.e.      see “Almost everywhere”
A.e. continuous function      211—212
a.s. = almost surely associated law      416—418
Abel limit      237
Abel summability      209 353 379
Abelian theorem      198—212
Abelian theorem for asymptotic balance      192
Abelian theorem for Borel summability      60
Abelian theorem for de Haan classes      159—165 172—174 189 191 242 281—283
Abelian theorem for differences      see “de Haan classes”
Abelian theorem for dominated variation      118—119
Abelian theorem for Fourier series and integrals      207—209
Abelian theorem for fractional integral      58
Abelian theorem for improper integral mean      200—201
Abelian theorem for improper Mellin convolution      202—203
Abelian theorem for indefinite integral      26—28 33—35 44 58—59 96—97 103—104 124 159—165
Abelian theorem for integral mean      198—201
Abelian theorem for integral of logarithm      31—32 165
Abelian theorem for Kohlbecker transform      257 281—283
Abelian theorem for LS transform      37—38 43—44 118—119 126 172—174 189 191 246
Abelian theorem for Mellin convolution      201—202
Abelian theorem for Mellin — Stieltjes convolution      209—212
Abelian theorem for O, o-versions of de Haan class      164—165 174
Abelian theorem for O-regular variation      96—97 124
Abelian theorem for power series      40
Abelian theorem for radial matrix transform      203—204 206 218—219 228—229
Abelian theorem for rapid variation      103—104 126
Abelian theorem for ratio      116—118 126
Abelian theorem for regular variation in general settings      425—426
Abelian theorem for slow variation with remainder      281—282
Abelian theorem for smooth variation      44
Abelian theorem for Stieltjes transform      40—41
Abelian theorem for Vuilleumier’s integral mean      200
Abelian theorem for Young conjugate      48—49
Abelian theorem in entire-function theory      303 305—310 318 320
Abelian theorem in probability theory      330—337
Abelian theorem of exponential type      247—257
Abel’s inequality      208
Additive function      1 4—5 “Hamel
Additive function in de Haan theory      140
Additive function in number theory      295
Additive notation      6 17 61 92
Additive-argument, holomorphic regularly varying function      425
Additive-argument, slowly varying function      81 124
Additive-argument, version of de Haan theory      129
Affine map      354—353 423
Age-dependent branching process      407 430
Alaoglu, L.      243
Algebra of regularly varying functions      47
Aljancic, S.      xvii
Almost decreasing      72
Almost increasing      72 123 183
Almost monotone      see “Almost decreasing” “Almost
Almost-sure limit set      420
Almost-sure limit theorem      404—405 415 419—420
Almost-uniform convergence      424—425
Amalgam (norm)      210—211 234
Analytic function      see “Entire function” “Holomorphic “Holomorphic
Analytic number theory      284—297 423—424
Anderson, C.W.      76
Anderson, J.M.      xix
Angular density (of zeros)      316—320
Aperiodic law      371
Aperiodic renewal sequence      369
Approximate convolution      see “Radial matrix” “Vuilleumier’s
Approximate regular variation      422
Approximation by regularly varying function      81—83 313
Arc-function      316
Arc-sine law      360—364 379 396—397
Asymmetric Cauchy law      see “Cauchy law”
Asymptotic balance      180—184 191—192
Asymptotic balance and Kohlbecker transform      283
Asymptotic balance and stochastic compactness of extremes      415
Asymptotic commutativity      224—225
Asymptotic equivalence and slow variation      14—15
Asymptotic equivalence and smooth variation      45
Asymptotic equivalence as equivalence relation      46—47
Asymptotic equivalence, Matuszewska’s      60
Asymptotic inverse      see “Inverse”
Asymptotically negligible array      339
Attraction      see “Domain of attraction”
Auxiliary function      see “Asymptotic balance” “The “Slow “Super-slow
Auxiliary function in de Haan theory      127—136 189
Auxiliary function in de Haan theory, asymptotic uniqueness      163
Auxiliary function in de Haan theory, bounded decrease      132—134
Auxiliary function in de Haan theory, bounded increase      129—132 150—151 153
Auxiliary function in de Haan theory, monotonicity      135
Auxiliary function in de Haan theory, O-regular variation      164—165
Auxiliary function in de Haan theory, positive decrease      130 134—135 152—153
Auxiliary function in de Haan theory, positive increase      136 152—153
Auxiliary function in de Haan theory, regular variation      127
Auxiliary function in de Haan theory, slow variation      145 164
Axiom of Choice      2
Baire analogue      see “Baire version”
Baire category      2
Baire extended regular variation (BER)      65—66
Baire function      2 61—62 123
Baire O-regular variation (BOR)      65—66
Baire property      2
Baire property, avoidance      11 18—19 134—136 140—145
Baire rapid variation      see “Rapid variation”
Baire regular variation (BR)      18;
Baire regular variation (BR), characterisation of limit      17 21
Baire set      see “Baire property”
Baire slow variation $(BR_{0})$       8
Baire slow variation $(BR_{0})$ , representation      15 21
Baire slow variation $(BR_{0})$ , uniform convergence      8—10 21
Baire version      8
Baire version of de Haan classes $(B\Pi_{g},BE\Pi_{g},BO\Pi_{g})$       128—129
Baire version of de Haan classes $(B\Pi_{g},BE\Pi_{g},BO\Pi_{g})$ , global indices      148
Baire version of de Haan classes $(B\Pi_{g},BE\Pi_{g},BO\Pi_{g})$ , local indices      146
Baire version of de Haan classes $(B\Pi_{g},BE\Pi_{g},BO\Pi_{g})$ , uniform convergence      140—141 144—145
Baire version of de Haan classes $(B\Pi_{g},BE\Pi_{g},BO\Pi_{g})$ , uniformity theorems      130—139
Baire-measurable      2
Baire’s theorem      5
Banach algebra      231—232 261—262 367—368
Banach space      213—215 243 438—439
Banach — Alaoglu theorem      243
Banach — Steinhaus theorem      214—215
Basis      5 10
Baumann, H.      229—230 234
Bekessy, A.      433
Bellman — Harris branching process      430
Bernoulli convergence      439
Bernoulli law      416
Bernstein’s Theorem      45
Berry — Esseen bounds      353
Bessel function      241
Beta integral      361 364
Beurling algebra      231—232
Beurling slow variation      76 120—122
Beurling, A.      120
Beurling’s generalised primes      see “Generalised primes”
Bienayme — Galton — Watson process      see “Simple branching process”
Biirmann — Lagrange theorem      434
Birth-and-death process      394
Bivariate domain of attraction      380 420
Bochner, S.      193
Bogenfunktion      316
Bojanic, R.      xvii 435
Borel measure      97
Borel sigma-algebra      436
Borel summability      60 353
Borel, E.      7—8
Bounded decrease (BD)      71—73 (see also “Dominated variation” “O-regular
Bounded decrease (BD) and integrals      94—97
Bounded decrease (BD) and O-version of Monotone Density Theorem      119—120
Bounded decrease (BD) and peaks      89 92—94

Bounded decrease (BD) and rapid variation      85—86 103
Bounded decrease (BD) as Tauberian condition      89 93—97 119—120 265 273—275
Bounded decrease (BD) of auxiliary function in de Haan theory J      32—34
Bounded decrease (BI) and integrals      94—97 124 126
Bounded decrease (BI) and inversion      124
Bounded decrease (BI) and O-version of Monotone Density Theorem      119—120
Bounded decrease (BI) and peaks      89 92—94
Bounded decrease (BI) and rapid variation      104
Bounded decrease (BI) as Tauberian condition      89 93—97; 274
Bounded decrease (BI) of auxiliary function in de Haan theory      129—132 150—151 153 185
Bounded decrease (BI), Tauberian theorem for      119—120 126
Bounded increase (BI)      71—73 (see also “Dominated variation” “O-regular
Bounded measure      437
Bounded variation (BV)      436—444 (see also “Locally bounded variation”)
bounds      see “Global bounds” “Local “Potter
Branch point      264
Branching process      336 397—408
Branching process, and functional equations      398—399 405 428
Branching process, classification      397—398
Branching process, extensions      406—408
Branching process, limit theorems      398—406
Brownian motion      340—341 358
Busy cycle      386
Busy period      386—388
c.f.      see “Characteristic function”
Canonical product      300—309 312 317 324
Canonical representation      74 154—158
Category (Baire)      2
Cauchy functional equation      2 4—5 428
Cauchy functional equation and characterisation of regular variation      17
Cauchy functional equation, avoidance      19
Cauchy law      347 350 359
Cauchy’s inequality      266
Cauchy’s integral formula      425
Cauchy’s Theorem      287
Central limit theorem      344 353
Central limit theory      339 344—347 350—354
Central limit theory for random numbers of r.v.s      395
Central limit theory, functional      341 353
Centring constants      327
Centring constants for extremes      408—411 413
Centring constants for sums      344—347 349 351—353
Centring constants in self-similarity      357—358
Centring constants, omission      347 349
Centring function      356—357 359
Cesaro density      397
Cesaro summability      19—20 59 246 262—263 353
CHARACTER      262
Characterisation in O, o-version of de Haan theory      170—171
Characterisation of de Haan class      189
Characterisation of equitightness      440—443
Characterisation of extended regular variation      67—68 73—74
Characterisation of global indices      149
Characterisation of Karamata indices      67—68 73—74
Characterisation of limit in de Haan theory      128 139—143 188—189
Characterisation of limit in Karamata theory      17—21 54—56
Characterisation of local indices      146—148 167—170 190
Characterisation of Matuszewska indices      68—75 125
Characterisation of O-regular variation      71—73
Characterisation of of slow variation      23—24 58
Characterisation of radial matrix      196—197 204—206
Characterisation of regular variation      259 264
Characterisation of self-similarity      355—356
Characterisation of the class $\Gamma$       191
Characteristic function (= Fourier — Stieltjes transform)      326 (see also “Law”)
Characteristic function (= Fourier — Stieltjes transform) and tail behaviour      336—337
Characteristic function (= Fourier — Stieltjes transform), compound Poisson type      339
Characteristic function (= Fourier — Stieltjes transform), entire      337
Characteristic function (= Fourier — Stieltjes transform), infinitely divisible      338—339
Characteristic function (= Fourier — Stieltjes transform), logarithm      338
Closure properties of regular variation      25—26;
Closure properties of slow variation      16
CLT      see “Central Limit Theorem”
Coin-tossing      372—373
Collective risk      see “Ruin”
Comparison theorem      see “Mercerian”
Complete asymmetry      396
Complete monotonicity      45
Complete randomness      416
Completely regular growth      316—321
Complex analysis      see “Entire function” “Holomorphic “Holomorphic
Complex-valued regular variation      423—424
Composition and variation measure      438
Composition of de Haan and regularly varying functions      189
Composition of functions in class $\Gamma$       180 191
Composition of regularly varying functions      26
Composition of slowly varying functions      16
Composition of smoothly varying functions      46—47
Compound geometric law      387—388
Compound Poisson law      338—339 345
Compound Poisson process      388—389
Concavity      45 430
Concentration function      395
Condition      see “Bounded decrease” “Bounded “Continuity “Darling “Dynkin “Kernel “Levin “Measurability “von “Positive “Positive “Right-continuous “Sinai’s “Slow “Slow “Spitzer’s “Stochastic “Tail “Tauberian “Wiener
Condition of limsuplimsup type      19 141—143
Condition of slow-decrease type      42—44 59 197—198 255
Condition of slow-increase type      19 141 146—148 255
Condition of slow-oscillation type      197—198 232
Condition, a.e. continuity      211—212
Condition, necessary for an implication      19 20 43—44 116 229 233—234
Conditionally convergent integral      see “Improper integral”
Conditionally convergent series      207—209 237—240
Conditioned limit theorem      394 398—405
Conjugate $\Pi$ -variation      189
Conjugate convex function      see “Young conjugate”
Conjugate index      48
Conjugate pair      29
Conjugate slowly varying function      see “de Bruijn conjugate”
Continuity in probability      see “Stochastic continuity”
Continuity interval      339—340 346
Continuity theorem for LS transform      38 116—117
Continuous interpolant      195 224—227
Continuous time, branching process      407
Continuous time, fluctuation theory      385
Continuous time, regenerative phenomenon      372 388
Conventions      see “Notation”
Conventions on $O\log O$       405
Conventions on empty set      66
Conventions on integrals      xix 33 194
Conventions on Landau’s symbol $\sim$       xix
Conventions on suprema and infima      66
Conventions on “Baire”      2
Conventions, finiteness of constants and indices      66 145
Conventions, terminological      xviii—xix
Convergence      see “Almost-sure limit theorems” “Narrow
Convergence in distribution      327
Convergence in law      327 423
Convergence in probability      372
Convergence, almost uniform      424—425
Convergence, finite-dimensional      356—357
Convergence-exponent      300 324
Converse Abelian theorem      212—217
Convexity      5 58 60 “Young
Convexity and indicator of entire function      315—316
Convexity and smooth variation      45
Convexity in branching processes      399 405
Convexity of Mellin transform of kernel      264 275
Convexity, geometry      316
Convolution      see “Lebesgue convolution” “Lebesgue “Mellin “Mellin
Convolution inequality      274—275 323
Convolution-power      326
Cosine series      see “Fourier series”
Counterexample      see “Examples” “Hamel
Counterexample for exceptional sets      115
Counterexample for Kohlbecker transform      282
Counterexample for monotone slow variation      57
Counterexample for O-regular variation      75
Counterexample for sequence formulation of regular variation      51
Counterexample for slow variation with rate      78 124
Counterexample for subexponentiality      430

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