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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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: Regular variation

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

:

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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$G_{\delta}$-set      62
$\emph{l}$-index      see g-index
a.e.      see Almost everywhere
A.e. continuous function      211212
a.s. = almost surely associated law      416418
Abel limit      237
Abel summability      209 353 379
Abelian theorem      198212
Abelian theorem for asymptotic balance      192
Abelian theorem for Borel summability      60
Abelian theorem for de Haan classes      159165 172174 189 191 242 281283
Abelian theorem for differences      see de Haan classes
Abelian theorem for dominated variation      118119
Abelian theorem for Fourier series and integrals      207209
Abelian theorem for fractional integral      58
Abelian theorem for improper integral mean      200201
Abelian theorem for improper Mellin convolution      202203
Abelian theorem for indefinite integral      2628 3335 44 5859 9697 103104 124 159165
Abelian theorem for integral mean      198201
Abelian theorem for integral of logarithm      3132 165
Abelian theorem for Kohlbecker transform      257 281283
Abelian theorem for LS transform      3738 4344 118119 126 172174 189 191 246
Abelian theorem for Mellin convolution      201202
Abelian theorem for Mellin Stieltjes convolution      209212
Abelian theorem for O, o-versions of de Haan class      164165 174
Abelian theorem for O-regular variation      9697 124
Abelian theorem for power series      40
Abelian theorem for radial matrix transform      203204 206 218219 228229
Abelian theorem for rapid variation      103104 126
Abelian theorem for ratio      116118 126
Abelian theorem for regular variation in general settings      425426
Abelian theorem for slow variation with remainder      281282
Abelian theorem for smooth variation      44
Abelian theorem for Stieltjes transform      4041
Abelian theorem for Vuilleumiers integral mean      200
Abelian theorem for Young conjugate      4849
Abelian theorem in entire-function theory      303 305310 318 320
Abelian theorem in probability theory      330337
Abelian theorem of exponential type      247257
Abels inequality      208
Additive function      1 45 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      354353 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      404405 415 419420
Almost-uniform convergence      424425
Amalgam (norm)      210211 234
Analytic function      see Entire function Holomorphic Holomorphic
Analytic number theory      284297 423424
Anderson, C.W.      76
Anderson, J.M.      xix
Angular density (of zeros)      316320
Aperiodic law      371
Aperiodic renewal sequence      369
Approximate convolution      see Radial matrix Vuilleumiers
Approximate regular variation      422
Approximation by regularly varying function      8183 313
Arc-function      316
Arc-sine law      360364 379 396397
Asymmetric Cauchy law      see Cauchy law
Asymptotic balance      180184 191192
Asymptotic balance and Kohlbecker transform      283
Asymptotic balance and stochastic compactness of extremes      415
Asymptotic commutativity      224225
Asymptotic equivalence and slow variation      1415
Asymptotic equivalence and smooth variation      45
Asymptotic equivalence as equivalence relation      4647
Asymptotic equivalence, Matuszewskas      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      127136 189
Auxiliary function in de Haan theory, asymptotic uniqueness      163
Auxiliary function in de Haan theory, bounded decrease      132134
Auxiliary function in de Haan theory, bounded increase      129132 150151 153
Auxiliary function in de Haan theory, monotonicity      135
Auxiliary function in de Haan theory, O-regular variation      164165
Auxiliary function in de Haan theory, positive decrease      130 134135 152153
Auxiliary function in de Haan theory, positive increase      136 152153
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)      6566
Baire function      2 6162 123
Baire O-regular variation (BOR)      6566
Baire property      2
Baire property, avoidance      11 1819 134136 140145
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      810 21
Baire version      8
Baire version of de Haan classes $(B\Pi_{g},BE\Pi_{g},BO\Pi_{g})$      128129
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      140141 144145
Baire version of de Haan classes $(B\Pi_{g},BE\Pi_{g},BO\Pi_{g})$, uniformity theorems      130139
Baire-measurable      2
Baires theorem      5
Banach algebra      231232 261262 367368
Banach space      213215 243 438439
Banach Alaoglu theorem      243
Banach Steinhaus theorem      214215
Basis      5 10
Baumann, H.      229230 234
Bekessy, A.      433
Bellman Harris branching process      430
Bernoulli convergence      439
Bernoulli law      416
Bernsteins Theorem      45
Berry Esseen bounds      353
Bessel function      241
Beta integral      361 364
Beurling algebra      231232
Beurling slow variation      76 120122
Beurling, A.      120
Beurlings 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.      78
Bounded decrease (BD)      7173 (see also Dominated variation O-regular
Bounded decrease (BD) and integrals      9497
Bounded decrease (BD) and O-version of Monotone Density Theorem      119120
Bounded decrease (BD) and peaks      89 9294
Bounded decrease (BD) and rapid variation      8586 103
Bounded decrease (BD) as Tauberian condition      89 9397 119120 265 273275
Bounded decrease (BD) of auxiliary function in de Haan theory J      3234
Bounded decrease (BI) and integrals      9497 124 126
Bounded decrease (BI) and inversion      124
Bounded decrease (BI) and O-version of Monotone Density Theorem      119120
Bounded decrease (BI) and peaks      89 9294
Bounded decrease (BI) and rapid variation      104
Bounded decrease (BI) as Tauberian condition      89 9397; 274
Bounded decrease (BI) of auxiliary function in de Haan theory      129132 150151 153 185
Bounded decrease (BI), Tauberian theorem for      119120 126
Bounded increase (BI)      7173 (see also Dominated variation O-regular
Bounded measure      437
Bounded variation (BV)      436444 (see also Locally bounded variation)
bounds      see Global bounds Local Potter
Branch point      264
Branching process      336 397408
Branching process, and functional equations      398399 405 428
Branching process, classification      397398
Branching process, extensions      406408
Branching process, limit theorems      398406
Brownian motion      340341 358
Busy cycle      386
Busy period      386388
c.f.      see Characteristic function
Canonical product      300309 312 317 324
Canonical representation      74 154158
Category (Baire)      2
Cauchy functional equation      2 45 428
Cauchy functional equation and characterisation of regular variation      17
Cauchy functional equation, avoidance      19
Cauchy law      347 350 359
Cauchys inequality      266
Cauchys integral formula      425
Cauchys Theorem      287
Central limit theorem      344 353
Central limit theory      339 344347 350354
Central limit theory for random numbers of r.v.s      395
Central limit theory, functional      341 353
Centring constants      327
Centring constants for extremes      408411 413
Centring constants for sums      344347 349 351353
Centring constants in self-similarity      357358
Centring constants, omission      347 349
Centring function      356357 359
Cesaro density      397
Cesaro summability      1920 59 246 262263 353
CHARACTER      262
Characterisation in O, o-version of de Haan theory      170171
Characterisation of de Haan class      189
Characterisation of equitightness      440443
Characterisation of extended regular variation      6768 7374
Characterisation of global indices      149
Characterisation of Karamata indices      6768 7374
Characterisation of limit in de Haan theory      128 139143 188189
Characterisation of limit in Karamata theory      1721 5456
Characterisation of local indices      146148 167170 190
Characterisation of Matuszewska indices      6875 125
Characterisation of O-regular variation      7173
Characterisation of of slow variation      2324 58
Characterisation of radial matrix      196197 204206
Characterisation of regular variation      259 264
Characterisation of self-similarity      355356
Characterisation of the class $\Gamma$      191
Characteristic function (= Fourier Stieltjes transform)      326 (see also Law)
Characteristic function (= Fourier Stieltjes transform) and tail behaviour      336337
Characteristic function (= Fourier Stieltjes transform), compound Poisson type      339
Characteristic function (= Fourier Stieltjes transform), entire      337
Characteristic function (= Fourier Stieltjes transform), infinitely divisible      338339
Characteristic function (= Fourier Stieltjes transform), logarithm      338
Closure properties of regular variation      2526;
Closure properties of slow variation      16
CLT      see Central Limit Theorem
Coin-tossing      372373
Collective risk      see Ruin
Comparison theorem      see Mercerian
Complete asymmetry      396
Complete monotonicity      45
Complete randomness      416
Completely regular growth      316321
Complex analysis      see Entire function Holomorphic Holomorphic
Complex-valued regular variation      423424
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      4647
Compound geometric law      387388
Compound Poisson law      338339 345
Compound Poisson process      388389
Concavity      45 430
Concentration function      395
Condition      see Bounded decrease Bounded Continuity Darling Dynkin Kernel Levin Measurability von Positive Positive Right-continuous Sinais Slow Slow Spitzers Stochastic Tail Tauberian Wiener
Condition of limsuplimsup type      19 141143
Condition of slow-decrease type      4244 59 197198 255
Condition of slow-increase type      19 141 146148 255
Condition of slow-oscillation type      197198 232
Condition, a.e. continuity      211212
Condition, necessary for an implication      19 20 4344 116 229 233234
Conditionally convergent integral      see Improper integral
Conditionally convergent series      207209 237240
Conditioned limit theorem      394 398405
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      339340 346
Continuity theorem for LS transform      38 116117
Continuous interpolant      195 224227
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 Landaus symbol $\sim$      xix
Conventions on suprema and infima      66
Conventions on Baire      2
Conventions, finiteness of constants and indices      66 145
Conventions, terminological      xviiixix
Convergence      see Almost-sure limit theorems Narrow
Convergence in distribution      327
Convergence in law      327 423
Convergence in probability      372
Convergence, almost uniform      424425
Convergence, finite-dimensional      356357
Convergence-exponent      300 324
Converse Abelian theorem      212217
Convexity      5 58 60 Young
Convexity and indicator of entire function      315316
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      274275 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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