Page 11 - Elements of Distribution Theory

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Contents ix

9.4 Watson’s Lemma 268
9.5 Laplace’s Method 276
9.6 Uniform Asymptotic Approximations 281
9.7 Approximation of Sums 288
9.8 Exercises 295
9.9 Suggestions for Further Reading 297
10. Orthogonal Polynomials 299
10.1 Introduction 299
10.2 General Systems of Orthogonal Polynomials 299
10.3 Classical Orthogonal Polynomials 312
10.4 Gaussian Quadrature 317
10.5 Exercises 319
10.6 Suggestions for Further Reading 321
11. Approximation of Probability Distributions 322
11.1 Introduction 322
11.2 Basic Properties of Convergence in Distribution 325
11.3 Convergence in Probability 338
11.4 Convergence in Distribution of Functions
of Random Vectors 346
11.5 Convergence of Expected Values 349
11.6 O p and o p Notation 354
11.7 Exercises 359
11.8 Suggestions for Further Reading 364
12. Central Limit Theorems 365
12.1 Introduction 365
12.2 Independent, Identically Distributed Random Variables 365
12.3 Triangular Arrays 367
12.4 Random Vectors 376
12.5 Random Variables with a Parametric Distribution 378
12.6 Dependent Random Variables 386
12.7 Exercises 395
12.8 Suggestions for Further Reading 399
13. Approximations to the Distributions of More
General Statistics 400
13.1 Introduction 400
13.2 Nonlinear Functions of Sample Means 400
13.3 Order Statistics 404
13.4 U-Statistics 415
13.5 Rank Statistics 422
13.6 Exercises 432
13.7 Suggestions for Further Reading 434
14. Higher-Order Asymptotic Approximations 435
14.1 Introduction 435
14.2 Edgeworth Series Approximations 435