INTRODUCTION 1
CHAPTER 1. WEAK CONVERGENCE IN METRIC SPACES 7
1. Measures in Metric Spaces, 7
Measures and Integrals, 7, Tightness, 9
2. Properties of Weak Convergence, 11
Portmanteau Theorem, 11, Other Criteria, 14
3. Some Special Cases, 17
Euclidean Space, 17, The Circle, 18, The
Space R™, 19, The Space C, 19, Product
Spaces, 20
4. Convergence in Distribution, 22
Random Elements, 22, Convergence in Distri-
bution, 23, Convergence in Probability, 24,
Product Spaces, 26
5. Weak Convergence and Mappings, 29
Continuous Mappings, 29, Main Theorem,
30, Integration to the Limit, 31, An Extension
of Theorem 5.1, 33
6. Prohorov's Theorem, 35
Relative Compactness, 35, The Direct Theorem,
37, The Converse, 40
Contents
7. First Applications, 41
Smooth Functions, 41, The Central Limit
Theorem, 42, Characteristic Functions, 45, The
Cramer-Wold Device, 48, Local and Integral
Limit Theorems, 49, Weak Convergence on the
Circle and Torus, 50
CHAPTER 2. THE SPACE С 54
8. Weak Convergence and Tightness in C, 54
Weak Convergence, 54, Tightness, 54, Random
Functions, 57, Coordinate Variables, 60
9. The Existence of Wiener Measure, 61
Wiener Measure, 61, The Brownian Bridge, 64,
Separable Stochastic Processes, 65
10. Donsker's Theorem, 68
The Theorem, 68, An Application, 70, A
Necessary Condition for Tightness, 73, Another
Proof of Donsker's Theorem, 73
11. Functions of Brownian Motion Paths, 77
Maximum and Minimum, 11, The Arc Sine Law,
80, The Brownian Bridge, 83
12. Fluctuations of Partial Sums, 87
Maxima, 87, Product Moments, 88, Applica-
tions, 89, Proof of Theorem 12.1, 91, Moments,
94, A Tightness Criterion, 95, Further In-
equalities, 98
13. Empirical Distribution Functions, 103
CHAPTER 3. THE SPACE D 109
14. The Geometry of D, 109
The Space D, 109, The Skorohod Topology,
111, Completeness ofD, 115, Compactness in D,
116, A Second Characterization of Compactness,
118, Finite-Dimensional Sets, 120
15. Weak Convergence and Tightness in D, 123
Finite-Dimensional Distributions, 123, Tight-
ness, 125, Random Elements of D, 128, A
Criterion for Convergence, 128, Criteria for
Existence, 130, Other Criteria, 133, Separable
Stochastic Processes, 134
16. Applications, 137
Donskefs Theorem, 137, Dominated Measures,
139, Empirical Distribution Functions, 141
17. Random Change of Time, 143
Randomly Selected Partial Sums, 143, Random
Change of Time, 144, Applications, 145,
Renewal Theory, 148
18. The Uniform Topology, 150
CHAPTER 4. DEPENDENT VARIABLES
19. Diffusion, 154
A Characterization of Brownian Motion, 154,
Other Diffusion Processes, 158
20. Mixing Processes, 166
y-Mixing, 166, Inequalities for Moments, 170,
Functional Central Limit Theorem, VIA, Inte-
grals in Place of Sums, 178, Nonstationarity,
179
21. Functions of Mixing Processes, 182
Preliminaries, 182, Functional Central Limit
Theorem, 184, Applications, 191, Diophantine
Approximation, 193, Nonstationarity, 194
22. Empirical Distribution Functions, 195
(p-Mixing Processes, 195, Functions of <p-
Mixing Processes, 199
23. Martingales, 205
24. Exchangeable Random Variables, 208
Sampling, 208, Exchangeable Variables, 212
APPENDIX I. METRIC SPACES
Separability, 215, Compactness, 217, Upper
Semicontinuity, 218, 77ie S/?aa? i?00, 218, 77ie
C, 220
APPENDIX II. MISCELLANY 222
Measurability, 222, Change of Variable, 222,
Tail Probabilities, 223, Scheffe's Theorem, 223,
Subspaces, 224, Product Spaces, 224, Measur-
ability of Dh, 225, Belly s Theorem, 226,
Kolmogorov's Theorem, 228, Measurability of
Some Mappings, 230, More Measurability, 232
APPENDIX III. THEORETICAL COMPLEMENTS 233
The Problem of Measure, 233, Separable
Measures, 234, The Topology of Weak Con-
vergence, 236, Prohorov's Theorem, 239
BIBLIOGRAPHY 243
SUMMARY OF NOTATION 248
INDEX 251