Suchen und Finden
Preface
5
Contents
8
Outline of Contents
16
Notation and Symbols
19
1 Introductory Measure Theory
22
1 Probability Theory: An Introduction
22
2 Basics from Measure Theory
23
3 The Probability Space
31
4 Independence; Conditional Probabilities
37
5 The Kolmogorov Zero-one Law
41
6 Problems
43
2 Random Variables
46
1 Definition and Basic Properties
46
2 Distributions
51
3 Random Vectors; Random Elements
64
4 Expectation; Definitions and Basics
67
5 Expectation; Convergence
75
6 Indefinite Expectations
79
7 A Change of Variables Formula
81
8 Moments, Mean, Variance
83
9 Product Spaces; Fubini’s Theorem
85
10 Independence
89
11 The Cantor Distribution
94
12 Tail Probabilities and Moments
95
13 Conditional Distributions
100
14 Distributions with Random Parameters
102
15 Sums of a Random Number of Random Variables
104
16 Random Walks; Renewal Theory
109
17 Extremes; Records
114
18 Borel-Cantelli Lemmas
117
19 A Convolution Table
134
20 Problems
135
3 Inequalities
139
1 Tail Probabilities Estimated via Moments
139
2 Moment Inequalities
147
3 Covariance; Correlation
150
4 Interlude on Lp-spaces
151
5 Convexity
152
6 Symmetrization
153
7 Probability Inequalities for Maxima
158
8 The Marcinkiewics-Zygmund Inequalities
166
9 Rosenthal’s Inequality
171
10 Problems
173
4 Characteristic Functions
176
1 Definition and Basics
176
2 Some Special Examples
185
3 Two Surprises
192
4 Refinements
194
5 Characteristic Functions of Random Vectors
199
6 The Cumulant Generating Function
203
7 The Probability Generating Function
205
8 The Moment Generating Function
208
9 Sums of a Random Number of Random Variables
211
10 The Moment Problem
213
11 Problems
216
5 Convergence
220
1 Definitions
221
2 Uniqueness
226
3 Relations Between Convergence Concepts
228
4 Uniform Integrability
233
5 Convergence of Moments
237
6 Distributional Convergence Revisited
244
7 A Subsequence Principle
248
8 Vague Convergence; Helly’s Theorem
249
9 Continuity Theorems
257
10 Convergence of Functions of Random Variables
262
11 Convergence of Sums of Sequences
266
12 Cauchy Convergence
275
13 Skorohod’s Representation Theorem
277
14 Problems
279
6 The Law of Large Numbers
284
1 Preliminaries
285
2 A Weak Law for Partial Maxima
288
3 The Weak Law of Large Numbers
289
4 A Weak Law Without Finite Mean
297
5 Convergence of Series
303
6 The Strong Law of Large Numbers
313
7 The Marcinkiewicz-Zygmund Strong Law
317
8 Randomly Indexed Sequences
320
9 Applications
324
10 Uniform Integrability; Moment Convergence
328
11 Complete Convergence
330
12 Some Additional Results and Remarks
334
13 Problems
342
7 The Central Limit Theorem
347
1 The i.i.d. Case
348
2 The Lindeberg-Levy-Feller Theorem
348
3 Anscombe’s Theorem
363
4 Applications
366
5 Uniform Integrability; Moment Convergence
370
6 Remainder Term Estimates
372
7 Some Additional Results and Remarks
380
8 Problems
394
8 The Law of the Iterated Logarithm
400
1 The Kolmogorov and Hartman-Wintner LILs
401
2 Exponential Bounds
402
3 Proof of the Hartman-Wintner Theorem
404
4 Proof of the Converse
413
5 The LIL for Subsequences
415
6 Cluster Sets
421
7 Some Additional Results and Remarks
429
8 Problems
437
9 Limit Theorems; Extensions and Generalizations
439
1 Stable Distributions
440
2 The Convergence to Types Theorem
443
3 Domains of Attraction
446
4 Infinitely Divisible Distributions
458
5 Sums of Dependent Random Variables
464
6 Convergence of Extremes
467
7 The Stein-Chen Method
475
8 Problems
480
10 Martingales
483
1 Conditional Expectation
484
2 Martingale Definitions
493
3 Examples
497
4 Orthogonality
503
5 Decompositions
505
6 Stopping Times
507
7 Doob’s Optional Sampling Theorem
511
8 Joining and Stopping Martingales
513
9 Martingale Inequalities
517
10 Convergence
524
11 The Martingale { E( Z | Fn)}
531
12 Regular Martingales and Submartingales
532
13 The Kolmogorov Zero-one Law
536
14 Stopped Random Walks
537
15 Regularity
547
16 Reversed Martingales and Submartingales
557
17 Problems
564
A Some Useful Mathematics
570
1 Taylor Expansion
570
2 Mill’s Ratio
573
3 Sums and Integrals
574
4 Sums and Products
575
5 Convexity; Clarkson’s Inequality
576
6 Convergence of (Weighted) Averages
579
7 Regularly and Slowly Varying Functions
581
8 Cauchy’s Functional Equation
583
9 Functions and Dense Sets
585
References
591
Index
603
Alle Preise verstehen sich inklusive der gesetzlichen MwSt.