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

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 494

Добавлена в каталог: 06.12.2009

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Предметный указатель

Random walk, randomly stopped      421
Random walk, symmetric      396
Rapid variation      83—88 (see also “The class $\Gamma$ ”)
Rapid variation and cumulative maximum      87
Rapid variation and inverse      87—88
Rapid variation and LS transform      126
Rapid variation and truncated moments      332—333
Rapid variation and uniformity theorems      122
Rapid variation of indefinite integral      103—104
Rapid variation, Abelian theorem      103—103 4 126
Rapid variation, analogue in de Haan theory      139
Rapid variation, Baire version      83—85
Rapid variation, characterisation      85—86
Rapid variation, counterexample      125
Rapid variation, examples      85—86
Rapid variation, Mercerian theorem      103—104
Rapid variation, monotone      193
Rapid variation, representation      86
Rapid variation, Tauberian theorem      124 126
Rapid variation, uniform convergence      83—85 139
Rapid variation, uniform versions      83
Rate of convergence (in probability theory) for extremes      413—414
Rate of convergence (in probability theory) in central limit theory      353
Rate of convergence (in probability theory) in fluctuation theory      379
Rate of convergence (in probability theory) in renewal theory      360 367—368 432
Rate of growth      see “Growth”
Rate of slow variation      see “Slow variation with remainder” “Super-slow
Ratio Tauberian theorem      116—118 126
Record time      415—416 418—419
Record value      415—418
Records      415—419
Recurrent event      see “Regenerative phenomenon”
Reductio ad absurdum      8
Regenerative phenomenon      369—372
Regenerative phenomenon and ladder epoch      378
Regenerative phenomenon, continuous-time version      372 388
Regular measure      437
Regular summation method      20 228
Regular variation (R)      see “Holomorphic regular variation” “Normalised “Regularly “Weak
Regular variation (R) and extended regular variation      71
Regular variation (R) and Karamata indices      67
Regular variation (R) and locally bounded variation      33
Regular variation (R) and proximate order      310—312
Regular variation (R) and subexponentiality      341 387—388 429—430
Regular variation (R) approximate      422
Regular variation (R) as Karamata case      128
Regular variation (R) in general settings      21 423—426
Regular variation (R) of denominator in de Haan theory      127
Regular variation (R) of derivative      39—40 42—43
Regular variation (R) of indefinite integral      26—28 30—31 33—35 39—40 42—43 58—59
Regular variation (R), at the origin      18
Regular variation (R), characterisation      259 264
Regular variation (R), characterisation of limit      16—17 54—56
Regular variation (R), closure properties      25—26
Regular variation (R), definition      18
Regular variation (R), elementary properties      26
Regular variation (R), history      xvii 18 20 311
Regular variation (R), index      18
Regular variation (R), notation      18 194
Regular variation (R), Potter bounds      25
Regular variation (R), representation      21 74
Regular variation (R), role in Abel — Tauber — Mercer theory      259
Regular variation (R), role in probability limit theorems      329 357—359
Regular variation (R), sequence formulation      50—51 60
Regular variation (R), uniform convergence      22—23
Regularly distributed zeros      320
Regularly varying function      see “Regularly varying sequence” “Regular
Regularly varying function and cumulative maximum      23—24 58
Regularly varying function as approximant      81—83 313
Regularly varying function as Stieltjes integrator      33—35
Regularly varying function, compositions of      26
Regularly varying function, growth      22
Regularly varying function, inverse      28—29
Regularly varying function, local boundedness      18
Regularly varying function, local integrability      18
Regularly varying function, monotone      54—57
Regularly varying sequence      52—53 56 60
Relative measure      see “Linear density”
Relative sequential compactness      439—441
Relative stability of maxima      415
Relative stability of records      418
Relative stability of sums      350 372—375
Renewal epoch      359
Renewal equation      367
Renewal function      359—369 373—374
Renewal function of ladder-height      380—382
Renewal process      359 368—369 387
Renewal sequence      369—372 375 432
Renewal theory      359—369 430 432
Renyi’s theorem      295
Representation      see “Onesided representation”
Representation and locally bounded variation      104
Representation and monotonicity      56—57
Representation canonical      74 154—158
Representation of de Haan classes      158—160 162—164
Representation of extended regular variation      74
Representation of extended Zygmund class      123
Representation of holomorphic regular variation      425
Representation of infinitely divisible law      339—340 343
Representation of near-monotonicity      108—109
Representation of O,o-versions of de Haan class      152—153
Representation of O-regular variation      74—76
Representation of quasi-monotonicity      106—109
Representation of rapid variation      86
Representation of regular variation      21 74
Representation of regularly varying sequence      53
Representation of self-controlled function      122
Representation of self-neglecting function      121—122
Representation of slow variation      12—16 57 104 122
Representation of slow variation with remainder      185—186
Representation of stable law      347—349
Representation of super-slow variation      187—188
Representation of the class $\Gamma$       178—179 191
Representation, extension by asymptotic equivalence      21
Representation, failure      21
Residual lifetime      359—364 373—374
Resnick, S.I.      xix
Reuter, G.E.H.      51
Riemann hypothesis      295
Riemann integrability      21 296
Riemann zeta-function and Lambert kernel      233 263 285—286
Riemann zeta-function and multiplicative functions      294
Riemann zeta-function in Prime Number Theorem      287—289 295
Riemann — Lebesgue lemma      241 263 277
Riemann — Stieltjes integral      438
Right-continuity in law      355
Right-continuous paths      340 355
rsc      see “Relative sequential compactness”
Ruin      389 421 431—432
s.s      see “Self-similarity”
Sample maximum      408—415
Scale of growth      15 (see also “Growth”)
Scaling property (of stable law)      343
Scheffe’s lemma      351—352
Schur, I.      214
Second-order theory      see “de Haan theory”
Section numbering      xviii
Selection principle      see “Helly’s selection principle”
Self-controlled function      122 186—188
Self-neglecting function      76 120—122
Self-neglecting function and extremes      412
Self-neglecting function and slow variation      126
Self-neglecting function and the class $\Gamma$       178—179
Self-neglecting function in Tauberian conditions      190—191
Self-similarity      354—359
Semi-continuity      48
Semigroup      see “Iteration of functions”
Semigroup in Steinhaus theory      4
Semigroup of affine maps      354
Semigroup of probability laws      372

Semigroup of renewal sequences      372
Semigroup of smoothly varying functions      46—47
Semistability      355
Seneta, E.      xvii 435
SEQUENCE      see “Regularly varying sequence”
Sequence formulation of regular variation      50—51 60
Sequence, Abelian theorem, converse Abelian theorem      see “Radial matrix”
Sequence, interpolant of      194—195 223
Sequence, null      214—215
Sequence, permanent      197 206 215—216
Sequence, radial      18
Sequence, Tauberian conditions for      197—198
Sequence, Tauberian theorem      see “Radial matrix”
Sequence, “majorisability”      167
Service backlog      388
Service load      388
Service time      385—388
Set indexing      414
Sevastyanov, B.A.      407
sgn      162
Side-condition      see “Condition” “Tauberian
Signed measure      437
Signed measure and radial matrix      223 228
Signed measure, variation norm      353
Simple branching process      397 428
Sinai’s condition      384—385
Sine series      see “Fourier series”
Single-server queue      see “Queue”
Sinusoidal indicator      314 320—321
Skewness parameter      347 380
Skip-free random walk      see “Left-continuous random walk”
Slack, R.S.      404 408
Slow decrease      41 (see also “Condition of slow-decrease type”)
Slow decrease and Cesaro convergence      19
Slow decrease and conditions of limsuplimsup type      19
Slow decrease and Frullani integral      35—36
Slow decrease and Karamata indices      67—68
Slow decrease and Mellin convolution      227
Slow decrease and radial matrix transform      226—228
Slow decrease and rapid variation      85—86
Slow decrease and Tauberian theorem for dominated variation      119
Slow decrease and “monotone density” representation of de Haan class      160
Slow decrease in Karamata Tauberian Theorem      43
Slow decrease in Monotone Density Theorem      42—43
Slow decrease in number theory      290
Slow decrease in Tauberian theory      19
Slow decrease, condition on sequences      197—198
Slow increase      41 (see also “Condition of slow-increase type”)
Slow increase and Karamata indices      67
Slow increase and Ratio Tauberian Theorem      116—118
Slow increase and slow decrease      19 43
Slow oscillation      41
Slow oscillation and radial matrix transform      223—225 228
Slow oscillation, condition on sequences      197
Slow variation $(R_{0})$       see “Beurling slow variation” “Normalised “Slowly
Slow variation $(R_{0})$ and de Haan classes      128 164
Slow variation $(R_{0})$ and gauge functions      110—110 12
Slow variation $(R_{0})$ and locally bounded variation      104
Slow variation $(R_{0})$ and near monotonicity      105—106 108—109
Slow variation $(R_{0})$ and quasi-monotonicity      109—110
Slow variation $(R_{0})$ and self-neglecting function      126
Slow variation $(R_{0})$ as Karamata case      128
Slow variation $(R_{0})$ at the origin      18
Slow variation $(R_{0})$ in general settings      423—426
Slow variation $(R_{0})$ of indefinite integral      26—28
Slow variation $(R_{0})$ , characterisation by monotonicity      23—24 58
Slow variation $(R_{0})$ , closure properties      16
Slow variation $(R_{0})$ , definition      6
Slow variation $(R_{0})$ , elementary properties      16
Slow variation $(R_{0})$ , examples      16
Slow variation $(R_{0})$ , notation      18
Slow variation $(R_{0})$ , off an exceptional set      113 115 125 406
Slow variation $(R_{0})$ , Potter bounds      25
Slow variation $(R_{0})$ , representation      12—14 104 122
Slow variation $(R_{0})$ , representation with specified properties      14—16 56—57
Slow variation $(R_{0})$ , uniform convergence      6—12
Slow variation with rate      see “Slow variation with remainder” “Super-slow
Slow variation with remainder      76—77 185—186
Slow variation with remainder and de Haan classes      192
Slow variation with remainder and Kohlbecker transform      247 281—282
Slowly oscillating function      see “Slow oscillation”
Slowly varying function      see “Additive-argument slowly varying function” “Slow
Slowly varying function, composition of      16
Slowly varying function, conjugate      see “Young conjugate”
Slowly varying function, convex equivalent      58
Slowly varying function, growth      16 79—81 124
Slowly varying function, integral of      27—28 30—31
Slowly varying function, local boundedness      13
Slowly varying function, local integrability      13
Slowly varying function, unbounded      58
Smith, R.L.      xix
Smooth variation      14 44—47 60
Smooth variation and holomorphic regular variation      313 425
Smooth variation and Mercerian theorems      282—283
Smooth variation and proximate order      311
Smooth variation and theorems of exponential type      255—256
Smooth version of de Haan class      165—167 282—283
Solidarity theorem      372
span      350—351 360
Sparre Andersen, E.      375
Spectral measure      see “Levy measure”
Spectral negativity      342 380 388
Spectral positivity      342 348—349
Spectral positivity and occupation time      396
Spectral positivity and tail behaviour      348—349
Spectral positivity in fluctuation theory      383—384
Spent lifetime      359—364 370—371 373—374
Spitzer, F.      375 386 397
Spitzer’s condition      379—384 396
St. Petersburg game      372
St. Petersburg paradox      372
Stability (for renewal sequence)      372
Stability (for sums)      see “Stable law”
Stable law      343—353 (see also “Cauchy law” “Positive “Spectral “Strict
Stable law, and occupation time      395—396
Stable law, and self-similarity      359
Stable law, and Spitzer’s condition      379—384
Stable law, characterisation      347
Stable law, domain of attraction      344—347
Stable law, in local limit theory      350—353
Stable law, in renewal theory      361 366
Stable law, index      347—350 380
Stable law, skewness parameter      347 380
Stable law, tail behaviour      347
Stable process      359 379 420
Stable queue      386—388
Stable subordinator      348 394—395
Stationary increments      340 389
Stationary sequence      420—421
Stationary transition probabilities      370
Stationary waiting-time      see “Waiting time”
Statistical applications      413—414 419 426
Steinhaus, H.      2—3 214—215
Stieltjes integrator      see “Lebesgue — Stieltjes integrator”
Stieltjes transform      40—41
Stirling’s formula      307—308 321
Stochastic compactness      375
Stochastic compactness of extremes      184 415
Stochastic compactness of occupation times      394
Stochastic compactness of sums      349 375
Stochastic continuity      340 355
Stochastic monotonicity      393
Stochastic process      328 340—341 “Brownian “Compound “Cumulative “Cumulative “Extremal “Gaussian “Levy “Markov “Markov “Point “Poisson “Random “Regenerative “Renewal “Self-similarity” “Stable “Subordinator”)
Stochastic process and narrow convergence      328—329
Stochastic process, finite-dimensional convergence      356—357
Stochastic process, finite-dimensional laws      328 354—357
Stochastic process, functional central limit theory      341 353
Stochastic process, independent increments      340 359 387 389
Stochastic process, marginal self-similarity      355—357
Stochastic process, Markov property      358 387

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