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

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Год издания: 1987

Количество страниц: 494

Добавлена в каталог: 06.12.2009

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Предметный указатель

Maximal term      298—299 324—325
Maximal type      see “Type (of entire function”
Maximum function      see “Cumulative maximum”
Maximum modulus      298—299
Maximum modulus and genus zero      304
Maximum modulus and holomorphic regular variation      313
Maximum modulus and minimum modulus      321—324
Maximum modulus and Nevanlinna characteristic      304—305
Maximum modulus and proximate order      311
Maximum modulus and zero-distribution      324—325
Maximum modulus, examples      321
Meagre      2
Mean lifetime      359 370
Mean lifetime, finite      360 365 367 370
Mean lifetime, infinite      360—367
Mean type      see “Type (of entire function”
Measurability      xix 437 “Non-measurable
Measurability in law      356
Measurability, avoidance      11 18—19 134—136 140—145 152—153
Measure      xix (see also “Lebesgue measure” “Lebesgue “Linear “Signed “Variation
Measure theory, difficulties      10
Measure theory, qualitative and quantitative aspects      7
Mejzler, D.G.      414
Mellin convolution      194
Mellin convolution and de Haan classes      242—247 260—263
Mellin convolution and entire functions      302—304 306—309 323
Mellin convolution in number theory      285 289—290
Mellin convolution, Abelian theorem      201—202 242
Mellin convolution, converse Abelian theorem      213—214
Mellin convolution, improper      202—203 243
Mellin convolution, Mercerian theorem      260—278
Mellin convolution, Tauberian theorem      221—222 277 230—231 243—247
Mellin transform      194
Mellin transform, improper      202
Mellin — Stieltjes convolution      194 443—444
Mellin — Stieltjes convolution in probability theory      327 393
Mellin — Stieltjes convolution, Abelian theorems      209—212 242—243
Mellin — Stieltjes convolution, reduction to Mellin convolution      209—210 237
Mellin — Stieltjes convolution, Tauberian theorem      231 234—237
Mellin — Stieltjes transform      205 327 393
Mercerian theorem      259—283
Mercerian theorem and Lambert transform      263
Mercerian theorem for Cesaro means      263
Mercerian theorem for de Haan classes      160—163 260—263 278—283
Mercerian theorem for differences      see “For de Haan classes”
Mercerian theorem for dominated variation      118—119
Mercerian theorem for Holder means      263
Mercerian theorem for indefinite integral      30—31 33—35 58—59 96—97 103—104 160—165
Mercerian theorem for integral of logarithm      31—32 165
Mercerian theorem for Kohlbecker transform      281—283
Mercerian theorem for LS transform      118—119 263 274 278—281
Mercerian theorem for Mellin convolution      260—278
Mercerian theorem for O,o-versions of de Haan class      164—165
Mercerian theorem for O-regular variation      96—97
Mercerian theorem for rapid variation      103—104
Mercerian theorem for slow variation with remainder      281—282
Mercerian theorem in entire-function theory      304—305
Mercer’s theorem      262
Meromorphic function      306
Mertens formulae      294 296
Method of moments      329
Method of moments in Darling — Kac theory      391—392 394—395
Method of moments in renewal theory      364
Method of monotone minorants      219—221
Minimal type      see “Type (of entire function”
Minimum modulus      321—324
Mittag — Leffler function      315 321 325 329
Mittag — Leffler law      329
Mittag — Leffler law and occupation times      391—397
Mittag — Leffler law and stable subordinator      349
Mittag — Leffler law, determined by its moments      391
Mittag — Leffler law, tail behaviour      337
Mixing      421
Mobius function      295
Mobius inversion formula      286
Mode, C.J.      407
Modified Schroder functional equation      398
Moment-generating function      337
Monotone density argument      334 364 371 387 396
Monotone equivalents in de Haan theory      159
Monotone equivalents in Karamata theory      see “Cumulative maximum function”
Monotone minorants      219—221
Monotone O-regularly varying function      65
Monotone rapidly varying function      103
Monotone regularly varying function      54—57 60
Monotone regularly varying sequence      56
Monotone slowly varying function      16—17
Monotonicity      see “Almost decreasing” “Almost “Cumulative “Near-monotonicity” “Quasi-monotonicity” “Zygmund
Monotonicity and Matuszewska index of inverse function      124
Monotonicity, as Tauberian condition      37—41 58—60 116—119 124—126 159—160 172—174 180 234—237 255—257
Monotonicity, characterising Karamata indices      68 73
Monotonicity, characterising local indices      167—170
Monotonicity, characterising slow variation      23—24 58
Multiplicative function      6 17 21
Multiplicative function in number theory      290—296 423—424
Multiplicative notation      6
Multitype branching process      407—408
Multivariate extremes      415
Multivariate regular variation      426
Narrow convergence      439—443
Narrow convergence and radiality      195—196 204—205 223—226
Narrow convergence in probability theory      328—329 443
Narrow convergence of renewal measure      363
Narrow topology      439
Near-monotonicity      104—106 108—111 210
Necessary      see “Condition necessary for an implication”
Nevanlinna characteristic      304—305
Non-decreasing asymptotic balance      see “Asymptotic balance”
Non-lattice law      345 350
Non-lattice law in Darling — Kac theory      395—396
Non-lattice law in local limit theory      351—353
Non-lattice law in renewal theory      360 365—367
Non-measurable function      10—11 61 81 122 146
Non-measurable set      4
Non-restrictive condition      see “Condition necessary for an implication”
Norm      see “Amalgam” “Supremum “Variation
Normal law and almost-sure limit theorems      420
Normal law and extremes      412
Normal law and infinite divisibility      339
Normal law and records      416—418
Normal law and stability      344 346—349
Normal law and tail behaviour      342 348
Normalised regular variation      44 58
Normalised regular variation and regularly varying sequences      53
Normalised regular variation, extension      123
Normalised slow variation      15
Normalised slow variation and near-monotonicity      109
Normalised slow variation and quasi-monotonicity      109
Normalised slow variation and Zygmund class      24—25
Normalised slow variation, extension      123
Normalised slow variation, locally bounded variation      104
Norming constants      327
Norming constants for extremes      408—411 413
Norming constants for sums      344—353
Norming constants in relative stability      372—374
Norming constants in self-similarity      357—358
Norming constants, regular variation      345 349 357—358
Norming function and occupation time      392—393
Norming function and stochastic monotonicity      393
Norming function in renewal theory      360
Norming function in self-similarity      356—359
Norming function, regular variation      358 392—393
Notation      see “Conventions”
Notation, additive      6 17 61 92;
Notation, multiplicative      6
Null sequence      214—215
Number theory      see “Analytic number theory”
Nyman, B.      231
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$       128—129 148—149
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ and global indices      149

O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ and indefinite integral      164—165
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ and indices transform      151
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ and LS transform      174
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ and slow variation with remainder      185
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ , Abelian theorem      164—165 174
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ , Baire version      148
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ , global bounds      171—172
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ , Mercerian theorem      164—165
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ , representation      152—153
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ , Tauberian theorem      174
O,o-versions of de Haan class $(O\Pi_{g},o\Pi_{g})$ , uniform convergence      133
O-analogue      see “O-version”
O-regular variation (OR)      65—66 (see also “Bounded decrease” “Bounded “Dominated
O-regular variation (OR) and differential equation      427
O-regular variation (OR) and indefinite integral      94—97 119—120 124—125
O-regular variation (OR) and integral of logarithm      165
O-regular variation (OR) and LS transform      118—119
O-regular variation (OR) and Matuszewska indices      123
O-regular variation (OR) and rapid variation      83
O-regular variation (OR) and theorems of exponential type      258
O-regular variation (OR) in branching processes      406
O-regular variation (OR) in Drasin — Shea theory      271 276
O-regular variation (OR) in renewal theory      365
O-regular variation (OR) of auxiliary function in de Haan class      164—165 181—183
O-regular variation (OR), Abelian theorem      96—97 124 165
O-regular variation (OR), as Karamata case      128
O-regular variation (OR), higher-dimensional analogue      426
O-regular variation (OR), Mercerian theorem      96—97 165
O-regular variation (OR), representation      74—76
O-regular variation (OR), Tauberian theorem      119—120 124
O-regular variation (OR), uniform convergence      65—66
O-verson      see “O of “O-regular
O-verson of Aljancic — Karamata theorem      165
O-verson of Monotone Density Theorem      119—120 126
O-verson of Vuilleumier — Baumann theory      230
Occupation time      368 389—397
Omey, E.      xix
One-sided peaks      88—93 95—96
One-sided representation      68 167—171
One-sided stable law      see “Positive stable law”
Order      see “Upper order”
Order (of canonical product)      300
Order (of entire function)      298—300 (see also “Proximate order”)
Order (of holomorphic function in a sector)      313
Order statistics      408 413 415
Ordinary differential equation      427
Ordinary Laplace transform      see “Laplace transform”
Oriented zeros      304
Orlicz space      66
P-function      372
p.g.f      see “Probability generating function”
Pakes, A.G.      388
Parseval identity      240
Partial attraction      349 420
Partial limit      181 183 204—204—5 270—271
Partial records      418
Partitions      284—287
Pathology      2 10—11
Peaks      see “One-sided peaks” “Polya
Penultimate approximation      413
Period (of renewal sequence)      369
permanence      197 206 215—216
Perron — Frobenius theory      407
Peter-and-Paul law      372—373
Pfluger, A.      319
Philipp, W.      421
Phragmen — Lindeloef indicator      see “Indicator”
Phragmen — Lindeloef principle      314
Pitt, H.R.      227
PNT      see “Prime Number Theorem”
Point process      341 414
Poisson process      341 387—388 416
Pole      264 274
Polya maximal means      74 154
Polya peaks      88—94
Polya peaks in Drasin — Shea theorem      268—270 273
Polya peaks in entire-function theory      322
Polya peaks in Jordan’s theorem      277
Polya’s extension of Dini’s theorem      55 60 175 411
Positive (definition)      xix
Positive decrease (PD)      71
Positive decrease (PD) and integrals      96—97 100
Positive decrease (PD) of auxiliary function in de Haan theory      130 134—135 152—153 183—184 185—186
Positive increase (PI)      71
Positive increase (PI) and integrals      94—96 98—99
Positive increase (PI) of auxiliary function in de Haan theory      136 152—153 183—184
Positive increase (PI) of inverse      124
Positive increase (PI), Tauberian theorem for      119—120
Positive stable law      349 361
Positive stable law and branching process      403
Positive stable law in fluctuation theory      380—382
Potter bounds      25 72—73 429
Power series      40
Preradial      see “Radial matrix”
Prerequisites      xviii
Prime numbers      290—297 (see also “Prime Number Theorem”)
Principle of uniform boundedness      214—215 440
Pringsheim, O.      264 266
probability density      see “Density”
Probability generating function      327
Probability integral transformation      416
Probability measure      see “Law”
Probability theory      326—422
Prohorov, Yu.V.      439
Proximate order      299—313
Proximate order and completely regular growth      316—321
Proximate order and indicator      313 315
Pure quasi-monotone      105
Quantifier in asymptotic balance      183
Quantifier in de Haan theory in limit theorems      139—143 163
Quantifier in de Haan theory in uniformity theorems      129—134 137
Quantifier in Karamata theory      see “Monotone regularly varying function” “Regularly “Sequence
Quantifier in Karamata theory in Characterisation Theorem      17 19
Quantifier in Karamata theory in uniformity theorems      61—65
Quasi-monotonicity      104—111
Quasi-monotonicity in Abelian theorem      200—203 207—209 240 337
Quasi-monotonicity in converse Abelian theorem      216—217
Quasi-monotonicity in probability theory      337
Quasi-radial      see “Radial matrix”
QUEUE      385—388 421 430
r.v.      see “Random variable”
Radial matrix (including preradial, quasi-radial)      195—197
Radial matrix (including preradial, quasi-radial) and Vuillemier — Baumann theory      229—230
Radial matrix (including preradial, quasi-radial) as approximate convolution      222—227
Radial matrix (including preradial, quasi-radial), Abelian theorem      203—204 206 218—219 228—229
Radial matrix (including preradial, quasi-radial), asymptotic commutativity      224—225
Radial matrix (including preradial, quasi-radial), characterisation      196—197 204—206
Radial matrix (including preradial, quasi-radial), converse Abelian theorem      215—216
Radial matrix (including preradial, quasi-radial), non-negative      219—221 225—229
Radial matrix (including preradial, quasi-radial), Tauberian theorem      217—221 228—229
Radial sequence      18
Radon measure      437
random records      418
Random set      341 371—372
Random signs      310
Random stopping      421
Random variable      326 (see also “Law”)
Random vector      328
Random walk      343 (see also “Domain of attraction for sum” “Left-continuous
Random walk and Brownian motion      358
Random walk and fluctuation theory      375—385
Random walk and local limit theory      350—353
Random walk and maxima      419—420
Random walk and queues      385—386
Random walk and relative stability      372
Random walk and renewal process      368—369
Random walk and stable laws      343 349
Random walk, asymptotic behaviour      376
Random walk, nonlinear norming      350
Random walk, occupation times      395—397

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