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Bingham N.H., Goldie C.M., Teugels J.L. — Regular variation
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Íàçâàíèå: Regular variationÀâòîðû: 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.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1987Êîëè÷åñòâî ñòðàíèö: 494Äîáàâëåíà â êàòàëîã: 06.12.2009Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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Ïðåäìåòíûé óêàçàòåëü
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 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 436 Locally bounded variation 104 Locally bounded variation 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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