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

Counterexample for uniformity in de Haan theory      135
Counterexample for uniformity in Karamata theory      63—64
Counterexample for “Karamata’s Theorem” for one-sided indices      95 99 102—103 125
Counterexample in Mercerian theorem for LS transform      281
Counterexample in Uniform Convergence Theorem      10—11 141
Cramer’s condition      354
Critical branching process      397 399—403
Croftian theorem      49—50
Crump — Mode — Jagers process      407
Csiszar, I.      134
Cumulant-generating function      404
Cumulative maximum function and characterisation of slow variation      58
Cumulative maximum function and inverse      124
Cumulative maximum function and rapid variation      87
Cumulative maximum function as monotone equivalent of regularly varying function      23—24
Cumulative maximum process      375—382 386
Cumulative minimum process      375—376
Cumulative sum process      see “Random walk” “Weak
Cuypers, J.      xix
D-K      see “Darling — Kac condition”
d.f.      see “Distribution function”
Dam      389
Darling — Kac condition (D-K)      390—394
Darling — Kac theory      see “Occupation times”
de Bruijn conjugate      29 78—79 189
de Bruijn conjugate and smooth variation      47
de Bruijn conjugate and tail behaviour of probability law      341—342
de Bruijn conjugate and Tauberian theorem of exponential type      252—253 257
de Bruijn conjugate and Young conjugate      47—49
de Bruijn conjugate, methods of calculation      78—79 433—435
de Haan classes $(\Pi_{g},\Pi)$       128 158—165 “Inversely “Smooth
de Haan classes $(\Pi_{g},\Pi)$ and extremes      410—410(11
de Haan classes $(\Pi_{g},\Pi)$ and indefinite integral      26 159—163
de Haan classes $(\Pi_{g},\Pi)$ and Kohlbecker transform      257 281—283
de Haan classes $(\Pi_{g},\Pi)$ and local indices      167—170
de Haan classes $(\Pi_{g},\Pi)$ and LS transform      172—174 189 191 278—281
de Haan classes $(\Pi_{g},\Pi)$ and Mellin convolution      242—247
de Haan classes $(\Pi_{g},\Pi)$ and slow variation      164
de Haan classes $(\Pi_{g},\Pi)$ and slow variation with remainder      185 192
de Haan classes $(\Pi_{g},\Pi)$ and tail of probability law      374
de Haan classes $(\Pi_{g},\Pi)$ in entire-function theory      306—308
de Haan classes $(\Pi_{g},\Pi)$ , Abelian theorem      159—163 172—174 189 191 242 281—283
de Haan classes $(\Pi_{g},\Pi)$ , characterisation of limit      128 139—143
de Haan classes $(\Pi_{g},\Pi)$ , composition with regularly varying function      189
de Haan classes $(\Pi_{g},\Pi)$ , conjugacy      189
de Haan classes $(\Pi_{g},\Pi)$ , higher-dimensional analogues      426
de Haan classes $(\Pi_{g},\Pi)$ , integral characterisation      189
de Haan classes $(\Pi_{g},\Pi)$ , Mercerian theorem      160—163 260—263 278—283
de Haan classes $(\Pi_{g},\Pi)$ , monotone density representation      159—160
de Haan classes $(\Pi_{g},\Pi)$ , representation      158—160 162—164
de Haan classes $(\Pi_{g},\Pi)$ , Tauberian theorem      159—160 162—163 172—174 189 191 243—247 281—283
de Haan classes $(\Pi_{g},\Pi)$ , uniform convergence      139
de Haan function      26 (see also “de Haan class”)
de Haan function, additive argument version      129
de Haan function, uniformity theorems      129—139
de Haan theory      xvii 127—192
de Haan, L.F.M.      xvii xix 88 145
de la Vallee Poussin, Ch.J.      287
Defective law      368 375
Degenerate law      see “Relative stability”
Degenerate law as arc-sine law      364;
Degenerate, notation      326
Delayed renewal process      360
Denominator in de Haan theory      see “Auxiliary function in de Haan theory”
Dense set      see “Quantifier”
density      see “Cesaro density” “Linear “Logarithmic
Density ( = probability density)      326 350—353 “Scheffe’s
Density ( = probability density) and convergence of maxima      411—413
Density ( = probability density) for smoothing      14 45 166
Density ( = probability density) of stable law      350
Density ( = probability density), unimodal      350 352
Dependent random variables      414 420—421
Derivative      see “Monotone Density Theorem”
Derivative as operator      47
Derivative of function in class $\Gamma$       180
Derivative of smoothly varying function      44—45 47
Difference set      3 7 9 57
Differential equation      427
Differentiating an asymptotic relation      113 (see also “Monotone Density Theorem”)
Diffusion process      394
Dini derivates      58 123
Dini’s theorem      55 60
Dirac measure      262
Dirichlet series      40 292 423—424
Discrete law      see “Lattice law”
Discrete regular variation      49—53
Discrete subexponentiality      343 431—432
Distribution function      326 (see also “Law”)
Domain of attraction for first-passage time      382
Domain of attraction for ladder epoch      380—384
Domain of attraction for ladder height      380 384—385
Domain of attraction for maximum      409—413 416—417
Domain of attraction for occupation time      396
Domain of attraction for records      416—418
Domain of attraction for renewal sequence      372
Domain of attraction for sum      344—348
Domain of attraction for sum and local limit theory      350—353
Domain of attraction for sum and maximum      419—420
Domain of attraction for sum and occupation time      396
Domain of attraction for sum and relative stability      350
Domain of attraction for sum and renewal process      369
Domain of attraction for sum and Spitzer’s condition      379—380 383—384
Domain of attraction for sum to positive stable law      349 361
Domain of normal attraction      348—349 382
Dominated variation      54 (see also “O-regular variation”)
Dominated variation and indefinite integral      98—103
Dominated variation and LS transform      118—119
Dominated variation and quasi-monotonicity      106—110
Dominated variation and ratio Tauberian theorems      116—118
Dominated variation and subexponentiality      429—431
Dominated variation, Abelian theorem      118—119
Dominated variation, counterexample      103
Dominated variation, Mercerian theorem      118—119
Dominated variation, Tauberian theorem      118—119 232
Double convolution      234
Drasin, D.      xix
Dual space      243 439
Dwass, M.      370
Dynkin — Lamperti condition      364—367
Dynkin — Lamperti problem      365
Edgeworth expansion      353
Egorov’s theorem      10
Elementary Tauberian theorems      217—222 229
Embrechts, P.A.L.      xix
Entire characteristic function      337
Entire function      298—325
Entire function of Valiron — Titchmarsh type      305 312
Entire function, completely regular growth      316—321
Entire function, examples      307—308; 315 321
Entire function, extremal properties      305 321 323—325
Entire function, indicator of      313—316
Entire function, minimum modulus      321—324
Entire function, negative zeros      301—308 324—325
Entire function, Nevanlinna characteristic      304—305
Entire function, oriented zeros      304
Entire function, proximate order      310—313
Entire function, real zeros      308—310
Entire function, regularly distributed zeros      320
Entire function, zero-distribution      301—313 316—321 324—325
Equitightness      439—443
Equitightness and radial matrix      196 204 223
Erdos, P.      134
Erickson, K.B.      371
Esseen, C.-G.      353
Estimation      see “Statistical applications”
Euler product for Gamma function      246 279
Euler product for multiplicative function      292
Euler product for Riemann zeta-function      286—287 294
Euler summability      353
Euler’s summation formula      289

Eventually of one sign      161—162
examples      see “Counterexample”
Examples de Bruijn conjugate      433—435
Examples, domain of attraction for extremes      412
Examples, entire functions      307—308 315 321
Examples, Peter-and-Paul law      372—374
Examples, quasi- and near-monotonicity      109—110
Examples, rapid variation      85—86 125
Examples, slowly varying functions      16
Examples, subexponentiality      430
Exceptional set      113—115 125
Exceptional set and branching processes      406
Exceptional set and completely regular growth      316—318 320
Exceptional set and convolution inequality      275 323—324
Exceptional set and entire function with negative zeros      304
Exceptional set in renewal theory      365—367 371
Excursion      395
Explosion      398 403—406
Explosive branching process      see “Infinite-mean branching process”
Exponential type      see “Type (of entire function”
Extended de Haan class $(E\Pi_{g})$       128 145—148
Extended de Haan class $(E\Pi_{g})$ and local indices      146
Extended de Haan class $(E\Pi_{g})$ , Baire version      146
Extended de Haan class $(E\Pi_{g})$ , canonical representation      154—158
Extended de Haan class $(E\Pi_{g})$ , representation      154
Extended de Haan class $(E\Pi_{g})$ , uniform convergence      137—138
Extended regular variation (ER)      65—66 74—75
Extended regular variation (ER) and Karamata indices      67 123
Extended regular variation (ER) and O-regular variation      71
Extended regular variation (ER) and regular variation      71
Extended regular variation (ER) and theorems of exponential type      258
Extended regular variation (ER) of integral      94—97
Extended regular variation (ER), canonical representation      74 157
Extended regular variation (ER), characterisation      73
Extended regular variation (ER), representation      74
Extended regular variation (ER), uniform convergence      66
External rate      76
Extinction      397—398 403
Extinction probability      398 405
Extinction time      397—398 403—404
Extremal domain of attraction      see “Domain of attraction for maxima”
Extremal law      408—414 416—418
Extremal process      414—415 420
Extremal properties (of entire functions)      305 321 323—325
Extremes      122 184 408—415
Faa di Bruno      60
Fatou’s Lemma      154 228
Feller, W.      xvii 237
Fenchel’s Duality Theorem      48
Finite dam      389
Finite-dimensional convergence      356—357
Finite-dimensional laws      328 354—357
First-passage time      382 (see also “Ladder epoch”)
Fluctuation theory      368 375—385
Fourier cosine transform      240—241
Fourier integral      209 240—241
Fourier inversion formula      240—241
Fourier series      see “Conditionally convergent series”
Fourier series, Abelian theorems      206—209
Fourier series, generalisations      207 237 242
Fourier series, integrability theorems      241—242
Fourier series, Tauberian theorems      232 237—240
Fourier sine-transform      207—209 240—241
Fourier — Stieltjes integral      209
Fourier — Stieltjes transform      see “Characteristic function”
Fourth proof of the UCT      9 62—63 84 134 137—138
Fractional integral      58 336
Frullani integral      35—36
Function algebra      see “Banach algebra” “Beurling
Functional equation      428 (see also “Cauchy functional equation” “Modified
Functional equation in de Haan theory      140
Functional equation in supercritical branching process      404
Functional limit theory (probability)      341 353 414 418—419
g-index      128 144—145
g-index and representation      158—160
Gamma class $(\Gamma)$       see “The class $\Gamma$ ”
Gamma distribution      412
Gamma function and Cesaro means      246
Gamma function and entire-function theory      307—308 320—321
Gamma function and Lambert summability      232—233
Gamma function and Mercerian theorem for LS transform      279
Gamma function and partitions      285—286
Gamma function, Euler product      246 279
Gaps in Karamata theory      127—128
Gauge function      110—113 208—210
Gaussian law      see “Normal law”
Gaussian process      420 422
Gegenbauer series      237
Gelfand theory      261—262
Generalised arc-sine law      see “Arc-sine law”
Generalised convolution      372
Generalised integers      295
Generalised inverse      see “Inverse”
Generalised primes      290 295
Generalised renewal theory      368
Generating function for partitions      286 (see also “Probability”)
Generating function genus      300—301 306—310 312 324
Generating function of renewal sequence      369
Gestrahlt      195
Gestrahlte Folgen      18
Gestrahlungsfunktion      195
GI/G/1      386—388
Global bounds in de Haan theory      130—139 171—172 189—190
Global indices      148—149 153
Gnedenko, B.V.      411
Group      see “Topological group”
Group in characterisation of limit in de Haan theory      140 142
Group in Steinhaus theory      4 20 57
Group of affine maps      354 423
Group of smoothly varying functions      46—47
Growth of entire function      300 310—313
Growth of regularly varying function      22
Growth of slowly varying function      16 79—81 124
Haar measure      194
Hadamard factorisation      300 304 319
Hadamard, J.      287
Hahn decomposition      437 439 441
Half-line      350 397
Hamel basis      5 10 57
Hamel pathology      5 21 64
Hank el transform      241
Hardy — Littlewood — Karamata theorem      xvii
Harmonic renewal function      384—385
Harmonic renewal theory      368
Harris, T.E.      407 430
Hazard function      410—411 415—418
Hazard rate      411—412
Heiberg, C.      84
Heine — Borel theorem      7—8
Helly’s selection principle and asymptotic balance      181
Helly’s selection principle and Drasin — Shea theorem      270
Helly’s selection principle and equitightness      440 442
Helly’s selection principle and occupation times      392 394
Helly’s selection principle in Ratio Tauberian Theorem      116
Heuristics, Abel — Tauber theorems      194
Heuristics, indefinite integral of regularly varying function      27
Higher-dimensional regular variation      426
Hincin, A Ya.      339 387
History, Abel — Tauber theory      193
History, regular variation      xvii 18 20 311
Holder means      247 263
Holomorphic function      see “Entire function”
Holomorphic function and de Bruijn conjugate      433—434
Holomorphic function and functional equations      428
Holomorphic function in a sector      312—317 424—425
Holomorphic regular variation (HR)      312—313 424—425
Homomorphism      355
Hopf, E.      377
Hunt process      340
i.d.      see “Infinite divisibility”

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