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

Idle period      386
Ikehara, S.      288
Immigration      406—407
Improper integral in Fourier integrals      207 240—241
Improper integral in integral means      200—201 216—217
Improper integral in Karamata’s theorem      33—34
Improper integral, notation      34
Improper Mellin convolution      202—203
Improper Mellin transform      202
Indefinite integral and de Haan classes      159—163
Indefinite integral and dominated variation      98—103
Indefinite integral and O, o-versions of de Haan class      164—165
Indefinite integral and O-regular variation      96—97 119—120 124
Indefinite integral and rapid variation      103—104
Indefinite integral, Abelian theorem      26—28 33—35 44 58—59 96—97 103—104 124 159—165
Indefinite integral, Mercerian theorem      30—31 33—35 58—59 96—97 103—104 160—165
Indefinite integral, slow variation of      26—28
Indefinite integral, smooth variation of      44
Indefinite integral, Tauberian theorem      39—40 42—44 58—59 119—120 124 159—163
Independent increments      340 359 387 389
INDEX      see “g-index” “Global “Karamata “Local “Matuszewska
Index for extended Zygmund class      123
Index of equivalence class of regularly varying functions      47
Index of Orlicz space      66
Index of rapid variation      83
Index of regular variation      18 67
Index of regularly varying sequence      52
Index of self-similar process      355—359
Index of smooth variation      44
Index of stable law      347—350 380
Index, conjugate      48
Indexes (division of material)      xviii—xix
Indicator (in entire-function theory)      313—318 320—321
Indicator diagram      316 319
Indices transform      150—151
Indices transform and one-sided representation      170—171
Indices transform and representation      157 158
Indices transform in asymptotic balance      184
Infinitary cases      146—147
Infinite dam      389
Infinite divisibility      337—343
Infinite divisibility and stable laws      343—344
Infinite divisibility and subexponentiality      430—431
Infinite divisibility of branching process limit      407
Infinite divisibility of maximum of random walk      379
Infinite divisibility of renewal sequence      372
Infinite divisibility, Levy — Hincin formula      339
Infinite divisibility, non-negativity      340 430—431
Infinite divisibility, tail behaviour      341—343 431
Infinite oscillation      16
Infinite-mean branching process      397 406
Infinitely divisible lattice law      342—343 432
Infinitesimal array      339
Ingham summability      290
Ingham’s method      288
Insurance      see “Ruin”
Integer part      8
Integrability theorems      241—242
Integrable function      437
integral      see “Improper integral” “Indefinite
Integral average      see “Integral mean”
Integral equation      261—262 271 278
Integral mean      see “Vuilleumier’s integral mean”
Integral mean, Abelian theorem      198—201
Integral mean, converse Abelian theorem      213
Integral transform      see “Characteristic function” “Fourier “Fourier “Fourier “Hankel “Lambert “Laplace “LS “Mellin “Stieltjes
Integral transform, convolution type      see “Mellin convolution” “Mellin
Integral transform, integrability theorems      242
Integral, as operator      47
Integral, convention      xix 33 194
Integral, fractional      58
Integral, Frullani      35—36
Integral, Lebesgue — Stieltjes      437—438
Integral, Stieltjes      see “Lebesgue — Stieltjes”
Integrating an asymptotic relation      see “Karamata’s Theorem”
Integration by parts      33 98 100 437—438
Inter-arrival time      385 387
Internal rate      76 186
Interpolant      194—195 223
Interval-radiality      see “Radial matrix”
Invariance principle      421
Invariant measure      406—407 428
Inverse      28—29
Inverse and asymptotic equivalence      60
Inverse and cumulative maximum      124
Inverse and de Bruijn conjugate      29
Inverse and smooth variation      46
Inverse of de Haan function      176—177
Inverse of function in the class $\Gamma$       176—177
Inverse of rapidly varying function      88
Inverse of regularly varying function      28—29
Inverse of slowly varying function      87—88
Inverse, calculation      78—79
Inversely asymptotic      190
Isometry      439
Iterates of logarithm      16 433
Iteration of functions      428
Ito representation      341
Jacobi series      207 237 242
Jagers, P.      407
Jensen’s formula      318
Jirina, M.      407
Jordan decomposition      107 437
Jordan, G.S.      xix
Jumps of Levy process      340—342
Kac, M.      389—395
Karamata case      128
Karamata indices      66—68
Karamata indices and rapid variation      83
Karamata indices and representations      74
Karamata indices in renewal theory      365
Karamata indices of integral      94—97
Karamata indices, characterisation      67—68 73—74 170
Karamata theory      xvii 1—128
Karamata, J.      xvii 122
Kernel      see “Integral transform”
Kernel condition      see “Wiener condition”
Kernel condition, a.e. continuity      211—212
Kernel condition, absolute continuity      209—210
Kernel condition, amalgam-norm condition      210—211 234
Kernel condition, continuity      234—237
Kernel condition, for Beurling algebra      231—232
Kernel condition, growth      210
Kernel condition, integrability      200—202 213—214
Kernel condition, non-negativity      222 230 245—246 263 265
Kernel, Bessel-function kernel      241
Kernel, Fourier kernel      241
Kernel, Wiener kernel      see “Wiener condition”
Kesten, H.      429—430
Key renewal theorem      367
Kohlbecker transform      49
Kohlbecker transform and Tauberian remainder theorems      247
Kohlbecker transform, Abel — Tauber — Mercer theorems      257 281—283
Kolmogorov, A.N.      310 398
Konig, H.      237
Korenblum, B.I.      231
Kronecker’s lemma      379
Kronecker’s theorem      54 142
kth records      418
Kwapien, S.      310
Ladder epoch      375
Ladder epoch, strict ascending      375—376 378 380—384
Ladder epoch, strict descending      383—384
Ladder epoch, weak ascending      388
Ladder epoch, weak descending      375—378 386
Ladder height      375
Ladder height, strict ascending      375—376 380—382 384—385
Ladder height, weak descending      375—377 386
Ladder step      375
Lagrange inversion      433

Lambert summability      232—233 288—290
Lambert transform      247 263 286
Lamperti, J.      355 364—365 381 401 411
Landau, E.      xvii
Landau’s symbols      xix
Laplace transform and additive-argument slowly varying function      81 124
Laplace transform and smooth variation      45
Laplace transform in Polya’s lemma      266—267
Laplace — Stieltjes transform      see “LS transform”
Laplace’s method      257—258
Large deviations      354 413
Lattice law      326—327 350
Lattice law and extremes      413—414
Lattice law in Darling — Kac theory      395—396
Lattice law in local limit theory      351—353
Lattice law in renewal theory      360 367 369—372
Lattice law, aperiodic      371
Law ( = probability law)      326 (see also “Arc-sine law” “Cauchy “Compound “Compound “Convergence “Degenerate “Density” “Extremal “Infinite “Lattice “Mittag “Narrow “Non-lattice “Normal “Stable “Symmetric “Tail “Type” “Variation
Law ( = probability law), absolutely continuous component      353
Law ( = probability law), aperiodic      371
Law ( = probability law), characteristic function      326
Law ( = probability law), defective      368 375
Law ( = probability law), determined by its moments      329
Law ( = probability law), finite-dimensional laws      354—357
Law ( = probability law), LS transform      326—327
Law ( = probability law), marginal      355—357
Law ( = probability law), moment-generating function      337
Law ( = probability law), Peter-and-Paul law      372—373;
Law ( = probability law), semigroup of      372
Law ( = probability law), singular component      353
Law ( = probability law), subordinated      368
Law ( = probability law), symmetrisation      344
Law ( = probability law), “regulairly varying moments”      335—336
Law of the iterated logarithm      415 420
Laws of large numbers      414—415 418
Lebesgue convolution      166 231 261 326
Lebesgue measure      xix
Lebesgue — Stieltjes convolution      326
Lebesgue — Stieltjes integral      437—438
Lebesgue — Stieltjes integrator      33 97 150 326
Lebesgue — Stieltjes measure      326
Left-continuous random walk      382—384 396
Leibniz’s Rule      266
Levin — Pfluger condition (LP)      319—320
Levin, B.Ja.      319
Levy measure      339—342 346 389
Levy process      340—342 385 388
Levy — Hincin formula      339—340 383
Lifetime      359—360
Lightbulb      359—360 368
LIL      see “Law of the iterated logarithm”
Limit function of radial matrix      195 204 223 225—226
Lindelof, E.L.      313—314
Linear density      113—115
Linear functional      214—215 438—439
Linear interpolant      223 225
Littlewood, J.E.      xvii
Local boundedness      13 (see also “Uniformity theorems” “Weak
Local boundedness and cumulative maximum      87
Local boundedness and subadditivity      123
Local boundedness in de Haan theory      130—131 133—134 136 144—145
Local boundedness of almost-increasing function      123
Local boundedness of rapidly varying function      84—85
Local boundedness of regularly varying function      18
Local boundedness of slowly varying function      13
Local indices      145—148
Local indices and indices transform      151
Local indices and representation      154—157
Local indices, Baire version      146
Local indices, characterisation      146—148 167—170 190
Local indices, formulae for      146—148 154 190
Local integrability of regularly varying function      18
Local integrability of slowly varying function      13
Local limit theory      350—353
Local limit theory and occupation times      395
Local limit theory and renewal theory      366
Local limit theory for extremes      413
Local limit theory in branching processes      404
Local uniformity      see “Polya’s extension of Dini’s theorem” “Uniform “Uniformity
Locally bounded variation $(BV_{loc})$       436
Locally bounded variation $(BV_{loc})$ of normalised slowly varying function      104
Locally bounded variation $(BV_{loc})$ of regularly varying function      33
Logarithm of characteristic function      338
Logarithmic density      115 125
Logarithmic densityand convolution inequalities      275 323—324
Logarithmic integral      287 295
Logarithmic order      313
Lognormal law      418
Lower order      73—74
Lower order (of entire function)      299 322 325
Lower semi-continuity      48
lp      see “Levin — Pfluger condition”
LS transform      37—39 (see also “Uniqueness theorem”)
LS transform and de Haan classes      172—174 189 191 263 278—281
LS transform and dominated variation      118—119
LS transform and irregular variation      118—119
LS transform and O,o-versions of de Haan class      174
LS transform and rapid variation      126
LS transform and truncated moments      333—335
LS transform and uniform distribution mod      1 296
LS transform of non-negative infinitely divisible law      340
LS transform of probability law      327
LS transform of renewal function      361
LS transform of stable law      348—349
LS transform, Abelian theorem      37—38 43—44 118—119 126 172—174 189 191
LS transform, bilateral      205
LS transform, continuity theorem      38 116—116
LS transform, kernel      233
LS transform, Mercercian theorem      118—119 263 274 278—281
LS transform, ratio Tauberian theorem      116—118
LS transform, Tauberian remainder theorem      247
LS transform, Tauberian theorem      37—38 43—44 59 118—119 172—174 189 191 233—234 237 246
LS transform, theorems of exponential type      247—254
Lusin’s theorem      439 441
m.s.s.      see “Marginal self-similarity”
M/G/1      387—389
Marginal law      355—357
Marginal self-similarity      355—357 409
Marginal type      355
Markov chain      368 370 372 395
Markov process      389—394
Markov property      358 387 390
Markov renewal process      368
Martingale      404
Matrix      see “Radial matrix” “Regular
Matuszewska indices      68—74 (see also “Bounded decrease” “Bounded “Dominated “O-regular “Positive “Positive
Matuszewska indices and almost-monotonicity      72
Matuszewska indices and entire-function theory      322
Matuszewska indices and indices of Orlicz spaces      66
Matuszewska indices and integrals      94—103 125
Matuszewska indices and O-regular variation      123
Matuszewska indices and one-sided peaks      89 92—93
Matuszewska indices and orders      74
Matuszewska indices and Polya peaks      89 93—94
Matuszewska indices and Potter-type bounds      72
Matuszewska indices and quasi-monotonicity      105
Matuszewska indices and rapid variation      83
Matuszewska indices and representation      74—76
Matuszewska indices and stochastic compactness of sums      375
Matuszewska indices in Drasin — Shea theory      269 273—274 276—277
Matuszewska indices in renewal theory      365
Matuszewska indices of auxiliary function in de Haan theory      128
Matuszewska indices of inverse function      124
Matuszewska indices, alternative definition      124
Matuszewska indices, characteriation      68—75 125
Matuszewska indices, counterexamples for      99 102—103 125
Matuszewska indices, finiteness      71—73
Maxima      see “Extremes”
Maxima and sum      419—420
Maximal correlation coefficient      421

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