Probability: A Graduate Course (eBook)

Allan Gut (Autor)

eBook Download: PDF

2006 | 2005
XXIV, 608 Seiten
Springer New York (Verlag)
978-0-387-27332-7 (ISBN)

Infos

This textbook on the theory of probability starts from the premise that rather than being a purely mathematical discipline, probability theory is an intimate companion of statistics. The book starts with the basic tools, and goes on to cover a number of subjects in detail, including chapters on inequalities, characteristic functions and convergence. This is followed by explanations of the three main subjects in probability: the law of large numbers, the central limit theorem, and the law of the iterated logarithm. After a discussion of generalizations and extensions, the book concludes with an extensive chapter on martingales.

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"e;I know it's trivial, but I have forgotten why"e;. This is a slightly exaggerated characterization of the unfortunate attitude of many mathematicians toward the surrounding world. The point of departure of this book is the opposite. This textbook on the theory of probability is aimed at graduate students, with the ideology that rather than being a purely mathematical discipline, probability theory is an intimate companion of statistics. The book starts with the basic tools, and goes on to chapters on inequalities, characteristic functions, convergence, followed by the three main subjects, the law of large numbers, the central limit theorem, and the law of the iterated logarithm. After a discussion of generalizations and extensions, the book concludes with an extensive chapter on martingales. The main feature of this book is the combination of rigor and detail. Instead of being sketchy and leaving lots of technicalities to be filled in by the reader or as easy exercises, a more solid foundation is obtained by providing more of those not so trivial matters and by integrating some of those not so simple exercises and problems into the body of text. Some results have been given more than one proof in order to illustrate the pros and cons of different approaches. On occasion we invite the reader to minor extensions, for which the proofs reduce to minor modifications of existing ones, with the aim of creating an atmosphere of a dialogue with the reader (instead of the more typical monologue), in order to put the reader in the position to approach any other text for which a solid probabilistic foundation is necessary. Allan Gut is a professor of Mathematical Statistics at Uppsala University, Uppsala, Sweden. He is the author of the Springer monograph "e;Stopped Random Walks"e; (1988), the Springer textbook "e;An Intermediate Course in Probability"e; (1995), and has published around 60 articles in probability theory. His interest in attracting a more general audience to the beautiful world of probability has been manifested in his Swedish popular science book Sant eller Sannolikt ("e;True or Probable"e;), Norstedts frlag (2002).From the reviews:"e;This is more substantial than the usual graduate course in probability; it contains many useful and interesting details that previously were scattered around the literature and gives clear evidence that the writer has a great deal of experience in the area."e;Short Book Reviews of the International Statistical Institute,  December 2005"e;...This book is a readable, comprehensive, and up-to-date introductory textbook to probability theory with emphasis on limit theorems for sums and extremes of random variables. The purchase is worth its price."e;Journal of the American Statistical Association, June 2006

Inhalt

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
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
4 Expectation Definitions and Basics
5 Expectation Convergence
6 Indefinite Expectations 79
7 A Change of Variables Formula 81
8 Moments, Mean, Variance 83
9 Product Spaces Fubini’s Theorem
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
17 Extremes Records
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
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
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
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
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
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
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

Produkt-Details

• Erscheint lt. Verlag: 16.3.2006
• Reihe/Serie: Springer Texts in Statistics
• Reihe/Serie: Springer Texts in Statistics
• Zusatzinfo: XXIV, 608 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Technik
• Schlagworte: central limit theorem • convergence • law of large numbers • law of the iterated logarithm • Martingales • Mathematical Statistics • measure theory • Probability Theory • Random Variable
• ISBN-10: 0-387-27332-8 / 0387273328
• ISBN-13: 978-0-387-27332-7 / 9780387273327