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Rachev S.T. — Probability Metrics and the Stability of Stochastic Models

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Название: Probability Metrics and the Stability of Stochastic Models

Автор: Rachev S.T.

Аннотация:

Concentrates on four specialized research directions as well as applications to different problems of probability theory. These include: description of the basic structure of p. metrics, analysis of the topologies in the space of probability measures generated by different types of p. metrics, characterization of the ideal metrics for the given problem and investigations of the main relationships between different types of p. metrics. The presentation here is given in a general form, although specific cases are considered as they arise in the process of finding supplementary bounds or in applications to important special cases.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 484

Добавлена в каталог: 19.05.2006

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Предметный указатель

$(\zeta_F)$ metric      36
$D|D|1|_{\infty}$ queueing model      230 233 238
$D|G|1|_{\infty}$ queueing model      230 237
-density (density of -distribution)      390
-distribution      388
$GI|GI|1|_{\infty}$ , system      226
$G|G|1|_{\infty}$ -system      223 225 226 237 238
-distance      28 106
-engineer’s metric      24 190
-metric      29 106
-metric (Kantorovich metric)      267
-metric between distribution functions      6 73
-probability compound metric      353
-version of       266
-version of stop-loss metric of order m $(d_{m,p})$       315
-difference pseudomoment      359
$ymmetric stable random variable      265 281
$ymmetry      7
$\Lambda$ -representation      68 69 70
$\Lambda$ -structure of probability semidistances      68 69 70
$\Lambda$ -structure of the Ky Fan metric and the Ky Fan distance      71
$\mu$ -lower bound with fixed difference of marginal moments      48 49
$\mu$ -upper bound with fixed marginal moments      47 49
$\mu$ -upper bound with fixed sum of marginal moments      47 49
$\mu$ -upper bound with marginal sum fixed      46 49
$\mu_{0,r}$ and $\nu_{0,r}$ -metrics (convolution metrics)      267
$\rho$ -uniform class      201
$\rho$ -uniform class (in a broad sense)      202
$\varepsilon$ -entropy of a measure      217
$\varepsilon$ -independence      230
(r, +)-ideal metric (simple)      374
(r, I)-ideal metrics (compound (r, +)-ideal and compound $(r,\vee)$ -ideal metrics)      381
(r, II)-ideal metrics (compound $(r, \vee)$ -ideal and simple (r, +)-ideal metrics)      381
(r, III)-ideal metrics) simple $(r, \vee)$ -ideal and simple (r, +)-ideal metrics      381
0-regularly varying function      349
Abel sum      355
Absolute pseudomoment of order s      146
Admissible plan      241
Aggregate claim in the individual model ( $S^{ind}$ )      313
Aging class of distributions      257 307
Ahiezer      428
Akaike      387
Akilov      3 425
Analytic (Souslin) set      451
Analytic space      13
Anastassiou      428 429
Appel      425
Arak      433
atom      17
Average compound distances      39
Average metric (first difference pseudomoment)      30
Bachi      425
Balance equation      241
Balinski      425 427
Balkema      372 414 432 433
Banach space of type p      374
Barlow      257
Barnes      425
Bartfai      426 427
Bartoszynski      424 428
Basu      428
Baxter      430
Bazaraa      93 108
Beasley      426
Beirlant      432
Bentkus      434 476
Berge      108 425
Bergstrom      431
Berk      434
Berkes      17 157 424 426
Bernstein      220
Bernstein CLT      220
Bernstein condition      218
Bernstein — Kantorovich functional limit theorem      218
Bernstein — Kantorovich invariance principle      218 220
Berry — Esseen theorem      281
Bertino      425 427
Beta density      390
Bhattacharya      75 289 429
Bickel      425 428
Billingsley      13 66 75 102 189 219 222 424 429
Birnbaum      9
Birnbaum — Orlicz (distance) space      9 10
Birnbaum — Orlicz distance      10
Birnbaum — Orlicz distance, (compound) ( $\Theta_H$ )      41
Birnbaum — Orlicz distance, (simple) ( $\theta_H$ )      31
Birnbaum — Orlicz uniform distance      32
Blackwell      15
Borel space      144
Borel-isomorphic spaces      74
Borovkov      226 228 429—432
Brown J.      426
Brunaldi      426
Cambanis      150 152 171 426 427
Cambanis — Simons — Stout formula      244
Cassandro      425
Cesaro sum      355 367
Characterization of a distribution (stability)      388
Chebyshev’s inequality      178
chi square ( $\chi^2$ ) distribution      261
chi square ( $\chi^2$ ) test of exponentiality      257
Chouila — Houri      108
Chow      228 377 384
Christoph      431
Claim sizes      299
Clopen set      15 16
Co-minimal distance      34
Co-minimal metric      32
Co-minimal metric, functional      33
Cohen      372
Cohn      13 14
Collective model ( $S^{coll}$ )      314
Collective model in risk theory      313
Compactness in the space of measures      130
Completed graph of a function      54
Completeness in the space of measures      130
Componentwise simple (K-simple) semimetric      147
Compound (r, $\vee$ )-ideal and simple (r, +)-ideal metrics, ((r, II)-ideal metrics)      381
Compound (r, +)-ideal and compound (r, $\vee$ )-ideal metrics ((r, I)-ideal metrics)      381
Compound distance      39
Compound max-ideal metric      380
Compound max-ideal metrics of order r >0      353
Compound metric      39
Compound Poisson distribution (approximation)      326
Compound Poisson model in risk theory      314
Compound Q-difference pseudomoment      79
Condition of limiting negligibility      218
Conjugate function      245
Consumption quality measure      241
Contamination by additive factor      262
Contamination by mixture      262
Convergence in the space of measures      130
Convolution metrics ( $\mu_{0,r}$ and $\nu_{0,r}$ -metrics)      267
Cost of transferring the mass      90
Covariance operator      372
Cramer      390 391 431
Csiszar      387 434
Cuesta      426 429
Czima      426
Daley      260
Dall’Aglio      97 427
Day      427
De Acosta      426—428
de Finetti’s theorem      400 401
de Finetti’s type characterization of scale mixtures of i.i.d. r.v.’s      403
de Haan      352 372 432 433
Deheuvels      433
Deland      426
Derings      425
DFR class of distributions      257

DFRA class of distributions      257
Diaconis      164 401 427 434
Difference pseudomoment      265 312 318 359
Discrete metric      148
Distance space      9
Dobrushin      160 249 425—428
Donsker-Prokhorov theorem      429
Doob’s inequality      437
Dotto      428
Doubly ideal metrics      373 374 381
Douglas      426
Dowson      426 429
Du Mouchel      432
Dual representation of the Kantorovich functional      106
Duality theorem      167
Dubinskaite      431
Dudley      3 6 13 14 17 75 78—80 105 108 109 113 115 119 155 158 164 217 423-^29
Dudley, metric      79
Dunford      3 9 110 114 258 317 424
D’Aristotile      164 427
Ellison      434
Engineer’s distance      24
Engineer’s metric      5 73
Epochs of claims      299
Equicontinuous class of functions      206
Factorial moments      334
Feller      390 411
Fernique      425 427
Final distribution of mass      90
First difference pseudomoment (average metric)      30
Forcina      425
Fortet      3 77 108 207 211 212 424 428 429
Fortet — Mourier metric      93
Frank      155
Frechet      423 425
Frechet, bounds      425
Freedman      164 401 425 427 428 434
Functional central limit theorem      218
G-weak convergence      76 204
G-weighted Prokhorov metric      71 204
Gaffke      426
Galambos      433
Galmacei      425
Gamma density      390
Gani      434
Garsia-Palomares      427
Gaussian measure      247
Gelbricht      426 429
Generalized Glivenko — Cantelli — Varadarajan theorem      215
Generalized inverse of a distribution function      29
Generalized Kantorovich functional      137
Generalized Kantorovich — Rubinstein functional      137
Generalized Wellner theorem      213
Genest      155
Geometric infinitely divisible distribution      415
Geometric maxima infinitely divisible (GMID) distribution      418 420
Gerber      432
Germaine      425
Ghouila-Houri      157 425
Gibbs random field      249
Gibson      426
Gikhman      221 429
Gine      376 427
Gini      96 97 425
Gini index of dissimilarity      96
givens      428 429
Glivenko — Cantelli — Varadarajan (generalized) theorem      215
Glivenko — Cantelli — Varadarajan theorem      113 158
GMID (geometric maxima infinitely divisible) distribution      418 420
Gnedenko      226 429
Goecke      425
Goldfarb      425
Gonzales      425
Goovaerts      432
gray      97 425 426
Greedy algorithm      425
Grinchevicius      432
Group of rigid motions on       223
H-average compound distance      39 180
H-metric      8
Hahn — Banach theorem      114
Hall      80 431 432
Hamming metric      97
Hampel      387
Haneveld      426
Hausdorff      13 424
Hausdorff, metric      8 13
Hausdorff, metric between bounded functions      54
Hausdorff, structure of a probability distance      60
Hausdorff, structure of the Prokhorov metric      60
Hazard rate function      308
Hellinger metric      423
Hennequin      210 423
Hilbert space      372
HNBUE class of distributions      257 260 261 262 307 310
HNBUE-distribution      416
HNWUE class of distributions      257 261 262
Hoeffding distribution      150
Hoeffding — Frechet inequality      172 340
Hoffman      425
Hoffman — Jorgensen      374 375 434
Hohlov      432
Homogeneity of order r      338
Huber      387 425
Ibragimov      432
Ideal (probability) metric      264
Ideal metrics      256
Identity property      7
IFR class of distributions      257 307
IFRA class of distributions      257 307
Iglehart      226 429
Ignatov      427 430—433
Individual model in risk theory      313
Infinite divisible distribution      412 415
Initial distribution of mass      90
Input flow      223
Insurance risk theory      299
Integral limit theorem for random motions      288
Inverse (generalized) of a distribution function      29
Isbell      426
Jarvis      93 108
Jensen’s Inequality      187
K-minimal metrics      141 146 147 164
K-simple (componentwise simple) semimetric      147
Kagan      434
Kakosjan      423 430 434
Kalashnikov      152 172 223 257 387 416 423 429 430 432 435
Kallenberg      208
Kantorovich      3 75 100 108 115 425—427 429
Kantorovich (generalized) functional      137
Kantorovich (multi-dimensional) problem      98
Kantorovich — Rubinstein (generalized) functional      137
Kantorovich — Rubinstein duality theorem      109
Kantorovich — Rubinstein functional      93 107
Kantorovich — Rubinstein problem      93
Kantorovich — Rubinstein — Kemperman problem of multistaged snipping      91
Kantorovich, distance ()      27 30
Kantorovich, functional      90 93 94 96 224
Kantorovich, metric      5 27 28 30 39 90 117 161 170 259 267
Kantorovich, theorem      105
Kantorovich’s mass transference problem      89
Karlin      194 428
Kaufman      14
Kellerer      141 246 425 427
Kelley      424
Kemperman      33 194 425 426 428
Kemperman geometric approach to moment problems      229 236
Kennedy      226 429
Kersten      208
Klebanov      434

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