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

Stochastic process, measurability in law      356
Stochastic process, right-continuity in law      355
Stochastic process, right-continuous paths      340 355
Stochastic process, stationary increments      340 389
Stochastic process, stochastic continuity      340 355
Stolz angle      425
Stopping time      421
Strict stability      343 359
Strong approximation (in probability)      419 421
Strong limit theorem      see “Almost-sure limit theorem”
Strong Markov process      340
Subadditivity      69 123
Subadditivity, pathology      5 146
Subadditivity, standard theory      123 261
Subcritical branching process      397—399 404 430
Subexponentiality      429—432 (see also “Discrete subexponentiality”)
Subexponentiality and infinite divisibility      341
Subexponentiality and queues      387—388
Subexponentiality and random stopping      421
Subexponentiality and regular variation      341 387—388 429—430
Subexponentiality and renewal theory      368
Submultiplicative function      64 261
Subordinated law      368 421
Subordinator      389 (see also “Stable subordinator”)
Subsequential limit      see “Partial limit”
Sum of random variables      see “Central limit theory” “Dependent “Random “Renewal “Weighted
Summability      see “Abel” “Borel” “Cesaro” “Euler” “Lambert
Summability in number theory      290
Summability in probability theory      353
Super-slow variation      76—77 186—188 413
Supercritical branching process      397 403—407
Supporting function      315—316
Supporting line      315
Supporting point      315
Supremum functional      379 385
Supremum norm      213 215 438
Symmetric Cauchy law      see “Cauchy law”
Symmetric law      344—345
Symmetric random walk      396
Symmetrisation      344
Symmetry      103 153 166
Tail balance      346—347 349 353
Tail behaviour      see “Domain of attraction” “Subexponentiality”
Tail behaviour and almost-sure limit theorems      420
Tail behaviour and characteristic function      336—337
Tail behaviour and de Haan classes      374
Tail behaviour and maxima and sums      419—420
Tail behaviour and random stopping      421
Tail behaviour and spectral positivity      348—349
Tail behaviour and truncated moments      330—335
Tail behaviour and truncated variance      330
Tail behaviour in branching processes      398—407
Tail behaviour of busy period      388
Tail behaviour of infinitely divisible law      341—343
Tail behaviour of ladder height      385
Tail behaviour of partial limit of occupation times      394
Tail behaviour of service time      386—388
Tail behaviour of stable law      347
Tail behaviour of waiting time      386—387
Tail behaviour, exponentially small tails      337 348—349
Tail behaviour, slowly varying tails      349—350 419
Tail-difference      336 (see also “Tail behaviour”)
Tail-sum      330 (see also “Tail behaviour”)
Takacs functional equation      388
Tauberian condition, discussion      19 43—44 147 167 191 193 217—218 229 265 “Bounded “Condition “Condition “Monotonicity “Slow “Slow
Tauberian remainder theorem      247
Tauberian theorem      see “Wiener’s second Tauberian theorem” “Wiener’s
Tauberian theorem and asymptotic balance      184 192
Tauberian theorem and self-neglecting functions      191
Tauberian theorem for Borel summability      60
Tauberian theorem for bounded increase      119—120 126
Tauberian theorem for Cesaro mean      59 246
Tauberian theorem for de Haan classes      159—163 172—174 189 191 243—247 281—283
Tauberian theorem for dominated variation      118—119 232
Tauberian theorem for Fourier sine and cosine series      232 237—240
Tauberian theorem for fractional integral      58;
Tauberian theorem for Holder means      247
Tauberian theorem for indefinite integral      38—40 42—44 58—59 119—120 124 159—163
Tauberian theorem for Kohlbecker transform      257 281—283
Tauberian theorem for Lambert summability      233 247
Tauberian theorem for LS transform      37—38 43—44 59 116—119 172—174 189 191 233—234 237 246
Tauberian theorem for Mellin convolution      221—222 227 230—231 243—247
Tauberian theorem for Mellin — Stieltjes convolution      231 234—237
Tauberian theorem for O,o-versions of de Haan class      174
Tauberian theorem for O-regular variation      119—120 124
Tauberian theorem for power series      40
Tauberian theorem for radial matrix transform      217—221 228—229
Tauberian theorem for regular variation in general settings      425—426
Tauberian theorem for slow variation with remainder      281—282
Tauberian theorem for Stieltjes transform      40—41
Tauberian theorem for the class $\Gamma$       192
Tauberian theorem for Vuilleumier’s, integral mean      231
Tauberian theorem for Young conjugate      48—49
Tauberian theorem in entire-function theory      303—310 318—320
Tauberian theorem in number theory      285—287 288—294 296—297
Tauberian theorem in probability theory      330—337
Tauberian theorem of exponential type      247—257 337
Tauberian theorem, complex      288 296—297
Tauberian theorem, elementary      217—222
Tauberian theorem, local      275
Tauberian theorem, ratio      116—118 126
Tauberian theorem, “general”      234—236
The class $\Gamma$       174—150 191
The class $\Gamma$ and extremes      410—411
Theorems of exponential type      247—258
Thomas — Fermi equation      427
Three series theorem      310
Tightness      439
Time-space point process      341 414
Titchmarsh, E.C.      305 312
Toeplitz — Schur theory      214
Toeplitz’s theorem      228
Tomic, S.      xvii
Topological group      423
Total variation      436—437

Traffic intensity      386—388
Transient renewal theory      368 430
Triangular array      339
Trigonometric convexity      313—314 316
Truncated moments      330—335
Truncated variance      330
Truncated variance and almost-sure limit theorem      420
Truncated variance and domain of attraction for sum      346—348
Truncated variance and stochastic compactness      375
Type (of entire function)      299 313—314 320
Type (of probability law)      327—328
Type (of probability law) and extremes      408—409
Type (of probability law) and records      416—417
Type (of probability law) and self-similarity      354—355
Type (of probability law) and stable law      343—344
UCT      see “Uniform Convergence Theorem”
Ultimate monotonicity      23—25
Ultraspherical series      237
Unbounded slowly varying function      58
Uniform boundedness principle      214—215
Uniform bounds      see “Global bounds” “Local “Potter
Uniform convergence for asymptotic balance      182—183
Uniform convergence for regular variation in general settings      423—425
Uniform convergence for slow variation with remainder      185
Uniform convergence for super-slow variation      187
Uniform convergence for the class $\Gamma$       175
Uniform convergence in de Haan theory      137—145
Uniform convergence in de Haan theory for extended de Haan class      137—138
Uniform convergence in de Haan theory for rapid-variation analogue      139
Uniform convergence in de Haan theory, for O,o-versions of de Haan class      133
Uniform convergence in de Haan theory, without measurability or Baire property      143—145
Uniform convergence in Karamata theory      6—12 22—23
Uniform convergence in Karamata theory and monotonicity      54—56
Uniform convergence in Karamata theory and rate of slow variation      76—77
Uniform convergence in Karamata theory for Baire versions      8—10 21 66 85
Uniform convergence in Karamata theory for Beurling slow variation      120—121
Uniform convergence in Karamata theory for extended regular variation      66
Uniform convergence in Karamata theory for O-regular variation      66
Uniform convergence in Karamata theory for rapid variation      83—85 124
Uniform convergence in Karamata theory for regular variation      22—23
Uniform convergence in Karamata theory for slow variation      6—12
Uniform convergence in Karamata theory, counterexample      10—11 141
Uniform convergence in Karamata theory, extension by asymptotic equivalence      21 145
Uniform convergence in Karamata theory, failure      10—11 21
Uniform convergence in Karamata theory, without measurability or Baire property      11—12
Uniform distribution mod 1      296—297
Uniform rapid variation      see “Rapid variation”
Uniformity theorems      see “Uniform convergence”
Uniformity theorems in de Haan theory      129—129 139 189
Uniformity theorems in Karamata theory      61—65
Unimodal density      350 352
Uniqueness theorem for Beurling algebra      231—232 237
Uniqueness theorem for LS transform      38 205 237 363
Upper end-point      410—412 414
Upper order      73—74 (see also “Order (of canonical product” “Entire “Holomorphic
Upper order and approximation by regularly varying function      81—83
Upper order and Drasin — Shea theorem      265 273
Upper order and exceptional sets      125
Upper order and Jordan’s theorem      275
Valiron — Titchmarsh type      305 312
Valiron, G.      xvii
Variation measure      436
Variation measure and gauge functions      111—112
Variation measure and quasi- and near-monotonicity      104—108
Variation norm      353 438—439 443—444
Vervaat, W.      xix
Very slow variation      11 (see also “Uniformity theorems”)
Virtual waiting-time      388
Vivanti — Pringsheim theorem      264 266
von Mangoldt function      288—290
von Mises conditions      411—413
Vuilleumier — Baumann theory      229—230
Vuilleumier, M.      167
Vuilleumier’s integral mean, Abelian theorem      200
Vuilleumier’s integral mean, converse Abelian theorem      216
Vuilleumier’s integral mean, Tauberian theorem      231
Waiting time      385—388 430
Weak convergence      see “Narrow convergence”
Weak dependence      420—421
Weak regular variation      19—20 21
Weak-star convergence      243 439
Weak-star topology      439
Weierstrass primary factor      300
Weighted function algebra      see “Beurling function algebra”
Weighted renewal theory      368
Weighted sum of random variables      352 374
Wiener condition in Tauberian theorem      228—230 232—233 243—245 290
Wiener condition, heuristics      194
Wiener process      see “Brownian motion”
Wiener Tauberian theory      227—237
Wiener — Hopf factorisation      377
Wiener — Ikehara theorem      288
Wiener, N.      234
Wiener’s second Tauberian theorem      234
Wiener’s Tauberian theorem      120 122 227 289
Wirsing, E.      424
Yaglom, A.M.      398
Yaglom’s critical limit theorem      403
Young conjugate      47—49 283
Zero distribution      301—313 316—321 324—325
Zero-counting function      see “Zero distribution”
Zeta function      see “Riemann zeta-function”
Zorn’s Lemma      5 10
Zygmund class      24—25
Zygmund class and Dini derivates      58
Zygmund class and regularly varying sequence      53
Zygmund class, extension      123
“Best-possible” theorem      212 214 216 229
“Double-sweep”      128 188
“General Tauberian theorem”      234
“Karamata’s Theorem” for one-sided indices      94—103
“Majorisability”      167 190
“Monotone-density” for de Haan classes      159—160;
“Monotone-density”, O-version      119—120 126
“Regularly varying moments”      335—336

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