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Bingham N.H., Goldie C.M., Teugels J.L. — Regular variation
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Íàçâàíèå: Regular variation
Àâòîðû: Bingham N.H., Goldie C.M., Teugels J.L.
Àííîòàöèÿ: 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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Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö
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Ãîä èçäàíèÿ: 1987
Êîëè÷åñòâî ñòðàíèö: 494
Äîáàâëåíà â êàòàëîã: 06.12.2009
Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
Ñêîïèðîâàòü ññûëêó äëÿ ôîðóìà | Ñêîïèðîâàòü ID
Ïðåäìåòíûé óêàçàòåëü
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 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 174—150 191
The class 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 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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