PREFACE TO THE CLASSICS EDITION ................................ xiii
PREFACE .......................................................... xv
LIST OF SYMBOLS ................................................ xvii
CHAPTER 1. WEAK CONVERGENCE OF PROBABILITY MEASURES AND
UNIFORMITY CLASSES ................................................ 1
1 Weak Convergence ............................................... 2
2 Uniformity Classes ............................................. 6
3 Inequalities for Integrals over Convex Shells ................. 23
Notes ......................................................... 38
CHAPTER 2. FOURIER TRANSFORMS AND EXPANSIONS OF CHARACTERISTIC
FUNCTIONS ........................................................ 39
4 The Fourier Transform ......................................... 39
5 The Fourier-Stieltjes Transform ............................... 42
6 Moments, Cumulants, and Normal Distribution ................... 44
7 The Polynomials  s and the Signed Measures Ps ................. 51
8 Approximation of Characteristic Functions of Normalized Sums
of Independent Random Vectors ................................. 57
9 Asymptotic Expansions of Derivatives of Characteristic
Functions ..................................................... 68
10 A Class of Kernels ............................................ 83
Notes ......................................................... 88
CHAPTER 3. BOUNDS FOR ERRORS OF NORMAL APPROXIMATION ............ 90
11 Smoothing Inequalities ........................................ 92
12 Berry-Esseen Theorem .......................................... 99
13 Rates of Convergence Assuming Finite Fourth Moments .......... 110
14 Truncation ................................................... 120
15 Main Theorems ................................................ 143
16 Normalization ................................................ 160
17 Some Applications ............................................ 164
18 Rates of Convergence under Finiteness of Second Moments ...... 180
Notes ........................................................ 185
CHAPTER 4. ASYMPTOTIC EXPANSIONS—NONLATTICE DISTRIBUTIONS ...... 188
19 Local Limit Theorems and Asymptotic Expansions for
Densities .................................................... 189
20 Asymptotic Expansions under Cramer's Condition ............... 207
Notes ........................................................ 221
CHAPTER 5. ASYMPTOTIC EXPANSIONS—LATTICE DISTRIBUTIONS ......... 223
21 Lattice Distributions ........................................ 223
22 Local Expansions ............................................. 230
23 Asymptotic Expansions of Distribution Functions .............. 237
Notes ........................................................ 241
CHAPTER 6. TWO RECENT IMPROVEMENTS ............................. 243
24 Another Smoothing Inequality ................................. 243
25 Asymptotic Expansions of Smooth Functions of Normalized
Sums ......................................................... 255
CHAPTER 7. AN APPLICATION OF STEIN'S METHOD ..................... 260
26 An Exposition of Gotze's Estimation of the Rate of
Convergence in the Multivariate Central Limit Theorem ........ 260
APPENDIX A.I. RANDOM VECTORS AND INDEPENDENCE .................. 285
APPENDIX A.2. FUNCTIONS OF BOUNDED VARIATION AND DISTRIBUTION
FUNCTIONS ........................................ 286
APPENDIX A.3. ABSOLUTELY CONTINUOUS. SINGULAR, AND DISCRETE
PROBABILITY MEASURES ............................. 294
APPENDIX A.4. THE EULER-MACLAURIN SUM FORMULA FOR FUNCTIONS
OF SEVERAL VARIABLES ............................. 296
REFERENCES ...................................................... 309
INDEX ........................................................... 315
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