Dembo A., Zeitouni O. — Large deviations techniques and applications :: Электронная библиотека попечительского совета мехмата МГУ

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Dembo A., Zeitouni O. — Large deviations techniques and applications

Dembo A., Zeitouni O. — Large deviations techniques and applications

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Название: Large deviations techniques and applications

Авторы: Dembo A., Zeitouni O.

Аннотация:

This book presents an introduction to the theory of large deviations. Large deviation estimates have proved to be the crucial tool required to handle many questions in statistics, engineering, statistial mechanics, and applied probability. The mathematics is rigorous and the applications come from a wide range of areas, including elecrical engineering and DNA sequences. The second edition includes new material on concentration inequalities and the metric and weak convergence approaches to large deviations. General statements and applications have been sharpened, new exercises added, and the bibliography updated. Amir Dembo is Associate Professor of Mathematics and Statistics at Stanford University, and Professor of Electrical Engineering at the Technion-Israel Institute of Technology. He currently serves on the editorial board of the Annals of Probability. Ofer Zeitouni is Professor of Electrical Engineering at the Technion-Israel Institute of Technology. He has served on the editorial board of the IEEE Transactions on Information Theory and currently serves on the editorial board of Stochastic Processes and Applications.

Язык:

Рубрика: Технология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1993

Количество страниц: 346

Добавлена в каталог: 24.09.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$B(\Sigma)$ -topology      239
$\delta$ -rate function      6
$\delta$ -smoothed rate function      283
$\mathcal{W}$ -topology      145 311
$\sigma$ -field      5 165 228 314
$\sigma$ -field, cylinder      239 288 292
$\tau$ -topology      236 239 246 260 264 278 288 292
Absolutely continuous functions      152 160 165 177
Absolutely continuous measures      314
Additive functional of Markov chain      see "Markov additive
Additive set function      165
Affine regularization      135
Algebraic dual      145—149 239 246 310
Alphabet      11 58 72 84
Approximate derivative      165
Arzela — Ascoli, theorem      158 316
Asymptotic expansion      99
Azencott      150 224
Bahadur      99 150 277
Baldi's theorem      139 244
Ball      308
Banach space      55 130 142 223 229 244 292 312
Bandwidth      217
Base of topology      106-110 248—251 255 307
Bayes      77 78
Berry — Esseen expansion      96 97
Bessel process      216
Bijection      111 308
Birkhoff's Ergodic Theorem      88
Borel $\sigma$ -field      5 102 165 237 274 314 318
Borel probability measures      144 151 228 247 281
Borel — Cantelli lemma      57 70
Bounded variation      165
Branching process      280
Bronsted — Rockafellar, theorem      142
Brownian motion      43 151 160—164 170 183 186 187—217 224 321
Brownian sheet      164—169 168
Bryc's theorem      124—129 253
Cauchy sequence      245 311
Central Limit Theorem (CLT)      34 96
Change of measure      33 53 223 236 278 293
Change point      151
Characteristic boundary      199—201 224
Chebycheff's bound      see "Inequalities Chebycheff"
Chernoff's bound      78
Closed set      307
Code      86—93
Coding gain      87
Compact set      7 8 38 73 127 131 308 309
Complete space      311
Concave function      42 133
Contraction principle      20 110 144 150 154 187
Contraction principle, approximate      117 150
Contraction principle, inverse      111 150
Controllability      199
Convergence      308
Convergence determining      298 320
Convex analysis      46 139 150 305—307
Convex hull      72 228 310
Countably additive      165
Covariance      94 168
Covering      308
Cramer's theorem      2 18—20 26—42 45 55 57 69 94 113 223 227—236 277
Cumulant generating function      26
CUSUM      225
Dawson — Gaertner, theorem      144 156 275
Decision test      see "Test"
Desire Andre      see "Reflection principle"
Diffusion process      187—188
Digital communication      46 53 169 225
Distortion      86 87 91
Divergence      238
DMPSK      169—175 224
DNA sequences      68 151
Doeeblin recurrence      279
Domain      4
Dominated convergence      317
Donsker      9 150
Doob      193 321
Duality lemma      134—138 240 306
Dynamical systems      151 200 280
Eberlein — Smulian, theorem      242 312
Effective domain      4
Elliptic      213
Ellis      9 303 see
Empirical mean      26 68 229
Empirical measure      3 12 21 64 81 236—303
Energy      74 291
Ensemble      90 291
entropy      13 92
Epigraph      139
Equicontinuous      169 316 320
Equilibrium point      196
Equivalent measures      76 314
Ergodic      61 87 88 92 260
Error probability      54 57 76 82 171 281—283
Essentially smooth      44 47 52 147 245 306
Euler — Lagrange      175
Exchangeable      74 85 291
Exit from a domain      196—213 224
Exponentially equivalent      114 151 153 166
Exponentially good approximations      115 117 119 175 189
Exponentially tight      8 39 49 106 111 123 127 136 139 148 150 229 236—237 251 286
Exposed hyperplane      44 50 139
Exposed point      44 50 139
Fatou's lemma      35 51 156 167 230 317
Feller continuous      267—268 279
Fenchel — Legendre transform      26 42 52 101 134—138 152 155 164 212 228 244 254 263—265
Field      165 280 314
Freidlin      3 9 187—195 222
Gaertner — Ellis, theorem      43 45—54 57 94 124 130 139—143 155 227 305
Gateaux differentiable      142 148 241
Gaussian process      53 164 168 280 303
Generator      119
Gibbs      3 57 72—75 280 291—302
Gibbs, measure      295 297 299
Gibbs, parameter      74
Girsanov      223
Green's function      198 213
Gronwall's lemma      188 191 193 208 210 221 322
Hahn — Banach theorem      137 306 310
Hahn's theorem      315
Hamilton — Jacobi      212
Hamiltonian      299
Hoeffding      81 83
Hoelder continuous      164
Homeomorphism      308
Hypercontractive      263 279
Hypermixing      see "Mixing"
Hypothesis testing      57 75—85 281—287 303
I continuity set      5
Inequalities, Azuma      36 55
Inequalities, Borell      164 169 235
Inequalities, Burkholder — Davis — Gundy      210 321
Inequalities, Cauchy — Schwartz      173 190
Inequalities, Chebycheff      30 33 36 38 42 45 49 132 133 141 158 163 193 205 238 244 267 300
Inequalities, Fernique      223
Inequalities, Hoeffding      35
Inequalities, Hoelder      28 37 263
Inequalities, isoperimetric      223
Inequalities, Jensen      13 19 28 40 78 93 178 240 241 243 262 268 293
Inequalities, Logarithmic Sobolev      279
Initial conditions      191 249 251
Initial measure      267
integral      314
Interacting particles      294 298 303
Interior      307
Invariant measure      265 269 270
Ioffe      306
Irreducible      58 60 63 65 71 250

Ito      193 321
k-scan process      67
Kofman      53
Kolmogorov's extension      274 276
Kullback — Leibler distance      238 278
Lanford      277 303
Langevin's equation      213
Laplace's method      120 150 277
Large deviation principle (LDP)      5—9
Large Deviation Principle (LDP) for Banach space      244 277
Large Deviation Principle (LDP) for continuous time Markov processes      119 246—247
Large Deviation Principle (LDP) for diffusion processes      187—195
Large Deviation Principle (LDP) for empirical mean      2 45 150
Large Deviation Principle (LDP) for empirical measure      3 16 61 68 237—239 278
Large Deviation Principle (LDP) for empirical process      273—277 275
Large Deviation Principle (LDP) for i.i.d. empirical sum      2 18 27 37 228
Large Deviation Principle (LDP) for Markov chains      2 58—68 60 246—253
Large Deviation Principle (LDP) for Markov occupation time      264—269
Large Deviation Principle (LDP) for multivariate random walk      164 165—169
Large Deviation Principle (LDP) for projective limits      143—149
Large Deviation Principle (LDP) for random walk      152—160
Large Deviation Principle (LDP) for sampling without replacement      23 289
Large Deviation Principle (LDP) for topological vector spaces      130—143
Large Deviation Principle (LDP), behavior under transformations      104 110—120
Large Deviation Principle (LDP), existence      106
Large Deviation Principle (LDP), uniqueness      103
Large exceedances      176—187 225
Lattice law      95—96
Law of Large Numbers      20 40 45 50 79 227
Lebesgue measure      165 168 252
Lebesgue's theorem      156 317
Level sets      37 122 144
Levy metric      237 243 247 319
Levy process      197 201
Likelihood ratio      76 81 176 281—282
Lipschitz continuous function      160 187—189 195—196 210 215 217 255 305
Log-likelihood ratio      76
Logarithmic moment generating function      26 40 44 69 88 95 130 228 230 265 277
Lower bound      6 7 39 45 84 125
Lower semicontinuous      4 42 103 121 308
Markov      see "LDP for Markov chain"
Markov additive functional      59 95 278
Markov chain      57 60 71 150 252 269 278
Markov kernel      269
Markov process, continuous time      119 151
Markov semigroup      9 261
Martingale      193 210 321
Maximal length segments      68
Mazur's theorem      229 313
Mean, of a Banach space valued variable      245
Measurable function      314
Measure space      314
Metric entropy      133 243—244
Metric space      4 9 102
Micro-canonical      291
Min-max theorem      42 133 181 186
Mixing      235 253—263 279 280
Mixing, $\psi$ -mixing      254 261
Mixing, hypermixing      261 263 264 267 279
Mogulskii      3 152 166 179
Monotone class theorem      268
Monotone convergence      317
Multi-index      164
Mutual information      87
Neighborhood      307
Neyman — Pearson      76—81 98 281—282
Noise      170 201
Non-coherent detection      46 53
Norm      311
Occupation time      264—269 277
Open set      307
Partially ordered set      143 309
Partition function      295 298
Peres      259 302
Perron — Frobenius      58 62
Phase Lock Loop      212 224
Pinsker      276 290
Pointwise convergence      144 153 156 166 268 289 306
Poisson process      163 197
Polish space      see "Topological Polish
Polygonal approximation and interpolation      153 166 184 220
Portmanteau theorem      238 274 320
Pre-compact set      105 133 164 285 308 316
Pre-exponent      57
Probability of error      see "Error probability"
Process level LDP      273—277
Product measure      57 75 91 93 113 228 292
Product topology      229 232 309
Prohorov's criterion      150 319
Projective limit      143—149 150 155 166 236 241 274 279 309
Projective topology      143—149 156 274 277 288 309
PuLSE      217—218
Quadrature noise      53
Quadrature signal      53 170
Quasi-potential      178 198 213
Queueing      201 225
Radar      213 218
Radon — Nikodym      92 163 165 238 293 299 316
Random variables, Bernoulli      35 73
Random variables, exponential      35 46 98 160
Random variables, Geometric      98
Random variables, Normal      2 35 40 46 52 94 113 161 183 186 217 219 262
Random variables, Poisson      35 163
Random walk      68 152—160
Random walk, multivariate      164—169
Range gate      218—219
Rate distortion      86—93 87 91
Rate function      4 7
Rate function, convex      27 37 109 131 148 229 237—243 254
Rate function, good      4 34 37 104 117 142 144 229 237—243 254 268 270
Refinements of LDP      94 225
Refinements of LDP of Bahadur and Rao      95 99
Reflection principle      163 321
Regular conditional probability distribution (r.c.p.d.)      272 318
Regular measure      315 319
Relative entropy      13 62 65 80 236 238 278
Relative interior      47 305
Relative topology      308
Right filtering set      143 309
Rockafellar's lemma      47 53
Ruelle      277 303
Sample path      150
Sampling without replacement      20—25 74 85 288—291
Sanov's theorem      12 16 18 36 41 61—64 73 83 227 236—246 277 287 293
Schilder's theorem      161 163 172 188 190 223 235
Separating space, set      310
Separating, $\ell$ -separated      261
Separating, well-separating      126
Sequentially compact      242 309
Set function      313
Set function, Additive, Countably additive      314
Shannon's Theorem      57 88
Shift invariant      65 271 275
Shrinks nicely      167 317
signal      53 169
Signal-to-noise ratio      53
Source coding      57 91
Spectral radius      57 63 67
Stationary      61 253 260 265
Statistical mechanics      see "Gibbs"
Steep      44 52 229 307
Stein's lemma      79
Stirling's approximation      14 22 55
Stochastic differential equation      187
Stochastic matrix      58
Strassen      223
Strictly convex      35
Strong solutions      322
Strong topology      307 see
Sub-additivity      37 93 150 227 231—232 235 246—253 257 277

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