This 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.
Author(s): Aad W. van der Vaart, Jon A. Wellner (auth.)
Series: Springer Series in Statistics
Edition: 1
Publisher: Springer-Verlag New York
Year: 1996
Language: English
Pages: 510
Tags: Probability Theory and Stochastic Processes
Front Matter....Pages i-xvi
Front Matter....Pages 1-1
Introduction....Pages 2-5
Outer Integrals and Measurable Majorants....Pages 6-15
Weak Convergence....Pages 16-28
Product Spaces....Pages 29-33
Spaces of Bounded Functions....Pages 34-42
Spaces of Locally Bounded Functions....Pages 43-44
The Ball Sigma-Field and Measurability of Suprema....Pages 45-48
Hilbert Spaces....Pages 49-51
Convergence: Almost Surely and in Probability....Pages 52-56
Convergence: Weak, Almost Uniform, and in Probability....Pages 57-66
Refinements....Pages 67-70
Uniformity and Metrization....Pages 71-74
Front Matter....Pages 79-79
Introduction....Pages 80-94
Maximal Inequalities and Covering Numbers....Pages 95-106
Symmetrization and Measurability....Pages 107-121
Glivenko-Cantelli Theorems....Pages 122-126
Donsker Theorems....Pages 127-133
Uniform Entropy Numbers....Pages 134-153
Bracketing Numbers....Pages 154-165
Uniformity in the Underlying Distribution....Pages 166-175
Front Matter....Pages 79-79
Multiplier Central Limit Theorems....Pages 176-189
Permanence of the Donsker Property....Pages 190-204
The Central Limit Theorem for Processes....Pages 205-224
Partial-Sum Processes....Pages 225-231
Other Donsker Classes....Pages 232-237
Tail Bounds....Pages 238-268
Front Matter....Pages 277-277
Introduction....Pages 278-283
M-Estimators....Pages 284-308
Z-Estimators....Pages 309-320
Rates of Convergence....Pages 321-338
Random Sample Size, Poissonization and Kac Processes....Pages 339-344
The Bootstrap....Pages 345-359
The Two-Sample Problem....Pages 360-366
Independence Empirical Processes....Pages 367-371
The Delta-Method....Pages 372-400
Contiguity....Pages 401-411
Convolution and Minimax Theorems....Pages 412-422
Back Matter....Pages 429-509
