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De LA Pena V., Gine E. — Decoupling: From Dependence to Independence: Randomly Stopped Processes U-Statistics and Processes Martingales and Beyond
De LA Pena V., Gine E. — Decoupling: From Dependence to Independence: Randomly Stopped Processes U-Statistics and Processes Martingales and Beyond

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Название: Decoupling: From Dependence to Independence: Randomly Stopped Processes U-Statistics and Processes Martingales and Beyond

Авторы: De LA Pena V., Gine E.


This book presents the theory and several applications of the decoupling princi-ple, which provides a general approach for handling complex problems involving dependent variables. Its main tools consist of inequalities used for breaking (decoupling) the dependence structure in a broad class of problems by introducing enough independence so that they can be analyzed by means of standard tools from the theory of independent random variables.
Since decoupling reduces problems on dependent variables to problems on related (conditionally) independent variables, we begin with the presentation of a series of results on sums of independent random variables and (infinite-dimensional) vectors, which will be useful for analyzing the decoupled problems and which at the same time are tools in developing the decoupling inequalities. These include several recent definitive results, such as an extension of Levy's maximal inequalities to independent and identically distributed but not necessarily symmetric random vectors, the Khinchin-Kahane inequality (Khinchin for random vectors) with best constants, and sharp decompositions of the Lp norm of a sum of independent random variables into functions that depend on their marginals only. A consequence of the latter consists of the first decoupling result we present, namely, comparing the Lp norms of sums of arbitrary positive random variables or of martingale differences with the Lp norms of sums of independent random variables with the same (one-dimensional) marginal distributions. With a few subjects, such as Hoffmann-J0rgensen's inequality, we compromise between sharpness and expediency and take a middle, practical road.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 392

Добавлена в каталог: 19.09.2006

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Предметный указатель
$\varepsilon$-capacity      216
(dual) bounded Lipschitz distance      180 214
Angularly symmetric distribution      268—269
Anscombe's Theorem for degenerate V-statistics      363 366
Anscombe's Theorem for sequences      362 363
Anscombe's Theorem for sums of independent variables      361
Asymptotic equicontinuity criterion      211
Banach space of type      2 249
Bracketing numbers      233
Burkholder — Davis — Gundy inequalities for martingales      34 316
Burkholder — Gundy inequalities for randomly stopped processes      81—82
Burkholder — Gundy type exponential inequality      84
Canonical kernel      137
Central limit theorem for B-valued V-statistics      250
Central limit theorem for martingales (Brown Eagleson)      330
Central limit theorem for two $\times$ two tables      331
Central limit theorem for U-processes      238—247
Central limit theorem for U-statistics      180—183
Chaining      216—218
CI condition      293
Completely degenerate kernel      see canonical kernel
Conditional Jensen inequalities for nonnegative, nondecreasing functions      123—125
Conditional Khinchin inequalities for nonnegati ve, nondecreasing functions      122—123
Conditionally symmetric sequences      293
Conditionally symmetric sequences, inequalities      302 308 312—315
Contraction principle      6—7 114
Convergence in law in $l^{\infty}(\mathcal{F}), l^{\infty}(T)$      209
Covering numbers (of a metric space)      216
Criterion function (for M-estimators)      265
Cumulative hazard function      281
Decoupled sequences      293
Decoupled version      293
Decoupling counter-examples      345 348
Decoupling of autorregressive models      295
Decoupling of constrained intersections of sets      298
Decoupling of constrained products      297
Decoupling of constrained sums      300
Decoupling of Gaussian chaos      183—186
Decoupling of intersections of tangent events      298 299
Decoupling of martingales      34—35 300 320 322
Decoupling of moment generating functions of tangent sequences      300
Decoupling of multilinear fonns in independent random variables      135—136
Decoupling of order statistics      299
Decoupling of products of tangent variables      297
Decoupling of randomly stopped Bessel processes      83
Decoupling of randomly stopped sums      54 62 321
Decoupling of randomly stopped U-statistics      352 356
Decoupling of sampling without replacement      295
Decoupling of sums of nonnegative variables      33—34
Decoupling of sums of tangent variables: $L_p$ moments      307 308 312 336 347 348
Decoupling of sums of tangent variables: moment generating function      300
Decoupling of sums of tangent variables: tail probabilities      302 308 313 345
Decoupling of U-processes: convex functions      107
Decoupling of U-processes: tail probabilities      125 134
Decoupling of U-statistics and U-processes      97—98
Decoupling of U-statistics: convex functions      99
Decoupling of U-statistics: moment generating functions      301
Decoupling of U-statistics: tail probabilities      125
Decoupling of UMD (Unifonn martingale difference)-martingales      323
Degenerate kernel      137
Degree of degeneracy of a U-statistic      137
Differentially subordinate sequences and inequalities      313—315
Dudley's theorem (on sample continuity of Gaussian processes)      219
Empirical process      237
Envelope of a class of functions      107 224
Exponential inequalities for canonical U-processes      252
Exponential inequalities for canonical U-statistics      167
Exponential inequalities for chaos      116 118
Exponential inequalities for martingales and ratios      369—370
Exponential inequalities for the ratio of a martingale over its conditional variance      369—374
Exponential inequalities for U-statistics      165 167 171
Exponential inequalities Hoeffding's, for U-statistics      165
Exponential inequalities, Bennett's      167
Exponential inequalities, Bennett's for martingales      367 368
Exponential inequalities, Bernstein's      166
Exponential inequalities, Bernstein's for martingales      367 368
Exponential inequalities, maximal, for processes (chaining)      215—221
Exponential inequalities, maximal, for random variables      189
Exponential inequalities, Prokhorov's for martingales      372
First passage times for a the maximum of a group of agents in a market      84
First passage times for Bessel processes      83 84
First passage times for sums of i.i.d. variables      54 55
First passage times for the maximum volume of a group of spheres      86 87
First passage times, comparison between two processes      92
Fubini's inequality for outer expectations      106
Gaussian chaos      117—118 120 122 173—180
Gaussian chaos process      180 220
Gaussian process      173 219
Generalized Minkowski inequality      112
Hermite polynomials      176
Hoeffding's decomposition      137
Hoeffding's inequality for sampling without replacement      295
Hoffmann — Jorgensen inequalities      8—15 47 155
Hoffmann — Jorgensen inequalities for U-statistics and processes      155—160
Hypercontractivity of Gaussian chaos      117—118
Hypercontractivity of multinomial linear forms      131—132
Hypercontractivity of Rademacher chaos      110—116
Hypergeometric distribution      331 332
Hypergeometric distribution, noncentral      331—333
Hypergeometric distribution, noncentral (representation as sum of independent variables)      333
Identifiability (of parameters)      265
Image admissible Suslin classes of functions      138
Integrability in the CLT      180 181
Integrability in the LIL      192 200—205
Integrability in the LLN      161—164
Isonormal Gaussian process      173
k-function      28—32 56 57
Khinchin inequalities      15—20 121—122
Khinchin — Kahane inequalities      15—20
L-function      35—42 57
Law of large numbers for B-valued U-statistics      234
Law of large numbers for decoupled U-processes      235
Law of large numbers for the empirical simplicial median      267
Law of large numbers for U-processes      228—233
Law of large numbers for U-statistics      160—164
Law of large numbers for V-processes      236
Law of the iterated logarithm for B-valued U-statistics, bounded      255
Law of the iterated logarithm for B-valued U-statistics, compact      262
Law of the iterated logarithm for decoupled and/or randomized U-statistics      198
Law of the iterated logarithm for U-processes, bounded      255
Law of the iterated logarithm for U-processes, compact      256
Law of the iterated logarithm for U-statistics, bounded      192—193
Law of the iterated logarithm for U-statistics, compact I      95
Levy's maximal inequalities      2—7
Levy's maximal inequalities for processes with independent increments      81
M-estimator      265 279
Marcinkiewicz inequalities      34
Marcinkiewicz law of large numbers      162
Marcinkiewicz type law of large numbers for U-statistics      235
Maximal inequality for exponential Orlicz norms      189
Measurability      8 15 106 138
Measurable classes of functions      138
Measurable envelope      107 224
Metric entropy      216
Newton's identities      175
Orlicz norms, Orlicz spaces      36
Outer integral, expectation, probability      106
P-Donsker class of functions      237
Packing numbers (of a metric space)      216
Paley — Zygmund argument      119
Polarization fonnula      174
Principle of conditioning, almost sure convergence      327
Principle of conditioning, weak convergence      328
Product limit estimator (Lynden — Bell)      282
Rademacher chaos      110—118 120 122—125
Rademacher chaos processes      220
Rademacher variables, Rademacher sequences      12 16
Random distances      228 231 241
Randomization in the law of the iterated logarithm for U-processes      148
Randomization inequalities for martingales      300
Randomization inequalities for sums      12 139
Randomization inequalities for U-statistics and U-processes: convex functions      140 144
Randomization inequalities for U-statistics and U-processes: tail probabilities      146 148
Rosenthal inequalities      43—46
Rosenthal inequalities for martingales      322
Rosenthal inequalities for martingales (sharp constants)      337
Sample bounded process      209
Sample continuous process      219
Sampling, conditionally independent      295
Sampling, conditionally independent, with replacement      295
Sampling, conditionally independent, without replacement      295
Separable process      218
Simplicial depth function      232 264
Simplicial depth process      232 248
Simplicial median, empirical simplicial median      265
Statistics of directions (example on)      248
Stochastic differentiability      274—275 280
Strassen's law of the iterated logarithm      194 205
Tangent sequences      293
Tetrahedral polynomial      118
Three series theorem      326
Truncated data      280—288
Two-$\times$-two tables      331
U-process      97 207
U-statistic      97
U-statistics, moments      358
U-statistics, randomization      358
U-statistics, randomly stopped (moments)      351 356 359
V-statistic or von Mises statistic      235
Vapnik — Cervonenkis class of sets      221
Vapnik — Cervonenkis subgraph class of functions      224
Wald's equations for randomly stopped processes      80
Wald's equations for sums of independent random variables      52 54
Wald's equations for U-statistics      351 356
Wald's equations, re-formulation      51
Weak convergence in probability      328
Woodroofe (Nelson — Aalen type) estimator      281
Young function or Young modulus      36 188
Young moduli of exponential type      188—189
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