## Weak Convergence and Empirical Processes: With Applications to StatisticsThis book tries to do three things. The first goal is to give an exposition of certain modes of stochastic convergence, in particular convergence in distribution. The classical theory of this subject was developed mostly in the 1950s and is well summarized in Billingsley (1968). During the last 15 years, the need for a more general theory allowing random elements that are not Borel measurable has become well established, particularly in developing the theory of empirical processes. Part 1 of the book, Stochastic Convergence, gives an exposition of such a theory following the ideas of J. Hoffmann-J!1Jrgensen and R. M. Dudley. A second goal is to use the weak convergence theory background devel oped in Part 1 to present an account of major components of the modern theory of empirical processes indexed by classes of sets and functions. The weak convergence theory developed in Part 1 is important for this, simply because the empirical processes studied in Part 2, Empirical Processes, are naturally viewed as taking values in nonseparable Banach spaces, even in the most elementary cases, and are typically not Borel measurable. Much of the theory presented in Part 2 has previously been scattered in the journal literature and has, as a result, been accessible only to a relatively small number of specialists. In view of the importance of this theory for statis tics, we hope that the presentation given here will make this theory more accessible to statisticians as well as to probabilists interested in statistical applications. |

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### Contents

Introduction | 2 |

Outer Integrals and Measurable Majorants | 6 |

Weak Convergence | 16 |

Product Spaces | 29 |

Spaces of Bounded Functions | 34 |

Spaces of Locally Bounded Functions | 43 |

The Ball SigmaField and Measurability of Suprema | 45 |

Hilbert Spaces | 49 |

2141 Finite Entropy Integrals | 239 |

2142 Uniformly Bounded Classes | 245 |

2143 Deviations from the Mean | 254 |

2144 Proof of Theorem 21413 | 257 |

Notes | 269 |

Statistical Applications | 277 |

Introduction | 278 |

MEstimators | 284 |

Convergence Almost Surely and in Probability | 52 |

Convergence Weak Almost Uniform and in Probability | 57 |

Refinements | 67 |

Uniformity and Metrization | 71 |

Notes | 75 |

Empirical Processes | 79 |

Introduction | 80 |

211 Overview of Chapters 23214 | 83 |

212 Asymptotic Equicontinuity | 89 |

213 Maximal Inequalities | 90 |

214 The Central Limit Theorem in Banach Spaces | 91 |

Maximal Inequalities and Covering Numbers | 95 |

221 SubGaussian Inequalities | 100 |

222 Bernsteins Inequality | 102 |

223 Tightness Under an Increment Bound | 104 |

Symmetrization and Measurability | 107 |

232 More Symmetrization | 111 |

233 Separable Versions | 115 |

GlivenkoCantelli Theorems | 122 |

Donsker Theorems | 127 |

252 Bracketing | 129 |

Uniform Entropy Numbers | 134 |

262 VCClasses of Functions | 141 |

263 Convex Hulls and VCHull Classes | 142 |

264 VCMajor Classes | 145 |

265 Examples and Permanence Properties | 146 |

Bracketing Numbers | 154 |

272 Monotone Functions | 159 |

273 Closed Convex Sets and Convex Functions | 162 |

274 Classes That Are Lipschitz in a Parameter | 164 |

Uniformity in the Underlying Distribution | 166 |

282 Donsker Theorems | 168 |

283 Central Limit Theorem Under Sequences | 173 |

Multiplier Central Limit Theorems | 176 |

Permanence of the Donsker Property | 190 |

2102 Lipschitz Transformations | 192 |

2103 Permanence of the Uniform Entropy Bound | 198 |

2104 Partitions of the Sample Space | 200 |

The Central Limit Theorem for Processes | 205 |

2112 Bracketing | 210 |

2113 Classes of Functions Changing with n | 220 |

PartialSum Processes | 225 |

2122 PartialSum Processes on Lattices | 228 |

Other Donsker Classes | 232 |

2132 Elliptical Classes | 233 |

2133 Classes of Sets | 236 |

Tail Bounds | 238 |

321 The Argmax Theorem | 285 |

322 Rate of Convergence | 289 |

323 Examples | 294 |

324 Linearization | 300 |

ZEstimators | 309 |

Rates of Convergence | 321 |

341 Maximum Likelihood | 326 |

342 Concave Parametrizations | 330 |

343 Least Squares Regression | 331 |

344 LeastAbsoluteDeviation Regression | 336 |

Random Sample Size Poissonization and Kac Processes | 339 |

352 Poissonization | 341 |

The Bootstrap | 345 |

362 The Exchangeable Bootstrap | 353 |

The TwoSample Problem | 360 |

371 Permutation Empirical Processes | 362 |

372 TwoSample Bootstrap | 365 |

Independence Empirical Processes | 367 |

The DeltaMethod | 372 |

392 Gaussian Limits | 376 |

393 The DeltaMethod for the Bootstrap | 377 |

394 Examples of the DeltaMethod | 381 |

310 Contiguity | 401 |

3101 The Empirical Process | 406 |

3102 ChangePoint Alternatives | 408 |

Convolution and Minimax Theorems | 412 |

3111 Efficiency of the Empirical Distribution | 420 |

Notes | 423 |

Appendix | 429 |

Inequalities | 430 |

Gaussian Processes | 437 |

A22 Exponential Bounds | 442 |

A23 Majorizing Measures | 445 |

A24 Further Results | 447 |

Rademacher Processes | 449 |

Isoperimetric Inequalities for Product Measures | 451 |

Some Limit Theorems | 456 |

More Inequalities | 459 |

A62 Multinomial Random Vectors | 462 |

A63 Rademacher Sums | 463 |

Notes | 465 |

467 | |

Author Index | 487 |

493 | |

506 | |

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Weak Convergence and Empirical Processes Aw Van Der Van Der Vaart,Jon Wellner No preview available - 2014 |

### Common terms and phrases

applied arbitrary assume assumption ball Banach space Borel measurable bounded bracketing central limit theorem Chapter collection compact complete Conclude condition consider consistent constant contained continuous converges converges to zero convex covering defined definition depending derivative differentiability distribution Donsker elements empirical process entropy envelope equal equicontinuity equivalent estimator Example exists expectation finite fixed follows functions Gaussian given gives Hence holds implies independent indexed inequality integral Lemma limit maximal maximum mean measurable measurable functions metric multiple norm normal numbers obtain outer partition points preceding probability probability measure Problem proof Proposition proved random variables range replaced respect sample satisfies semimetric separable sequence shows side space statistics subset sufficiently Suppose surely term theorem theory true uniform uniformly values vector weak convergence yields