Gut Probability: A Graduate Course

1. Auflage 2006
ISBN: 978-0-387-27332-7
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 608 Seiten

Reihe: Springer Texts in Statistics

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.

Autoren/Hrsg.

Gut, Allan

Inhaltsverzeichnis

1;Preface;5
2;Contents;8
3;Outline of Contents;16
4;Notation and Symbols;19
5;1 Introductory Measure Theory;22
5.1;1 Probability Theory: An Introduction;22
5.2;2 Basics from Measure Theory;23
5.3;3 The Probability Space;31
5.4;4 Independence; Conditional Probabilities;37
5.5;5 The Kolmogorov Zero-one Law;41
5.6;6 Problems;43
6;2 Random Variables;46
6.1;1 Definition and Basic Properties;46
6.2;2 Distributions;51
6.3;3 Random Vectors; Random Elements;64
6.4;4 Expectation; Definitions and Basics;67
6.5;5 Expectation; Convergence;75
6.6;6 Indefinite Expectations;79
6.7;7 A Change of Variables Formula;81
6.8;8 Moments, Mean, Variance;83
6.9;9 Product Spaces; Fubini’s Theorem;85
6.10;10 Independence;89
6.11;11 The Cantor Distribution;94
6.12;12 Tail Probabilities and Moments;95
6.13;13 Conditional Distributions;100
6.14;14 Distributions with Random Parameters;102
6.15;15 Sums of a Random Number of Random Variables;104
6.16;16 Random Walks; Renewal Theory;109
6.17;17 Extremes; Records;114
6.18;18 Borel-Cantelli Lemmas;117
6.19;19 A Convolution Table;134
6.20;20 Problems;135
7;3 Inequalities;139
7.1;1 Tail Probabilities Estimated via Moments;139
7.2;2 Moment Inequalities;147
7.3;3 Covariance; Correlation;150
7.4;4 Interlude on Lp-spaces;151
7.5;5 Convexity;152
7.6;6 Symmetrization;153
7.7;7 Probability Inequalities for Maxima;158
7.8;8 The Marcinkiewics-Zygmund Inequalities;166
7.9;9 Rosenthal’s Inequality;171
7.10;10 Problems;173
8;4 Characteristic Functions;176
8.1;1 Definition and Basics;176
8.2;2 Some Special Examples;185
8.3;3 Two Surprises;192
8.4;4 Refinements;194
8.5;5 Characteristic Functions of Random Vectors;199
8.6;6 The Cumulant Generating Function;203
8.7;7 The Probability Generating Function;205
8.8;8 The Moment Generating Function;208
8.9;9 Sums of a Random Number of Random Variables;211
8.10;10 The Moment Problem;213
8.11;11 Problems;216
9;5 Convergence;220
9.1;1 Definitions;221
9.2;2 Uniqueness;226
9.3;3 Relations Between Convergence Concepts;228
9.4;4 Uniform Integrability;233
9.5;5 Convergence of Moments;237
9.6;6 Distributional Convergence Revisited;244
9.7;7 A Subsequence Principle;248
9.8;8 Vague Convergence; Helly’s Theorem;249
9.9;9 Continuity Theorems;257
9.10;10 Convergence of Functions of Random Variables;262
9.11;11 Convergence of Sums of Sequences;266
9.12;12 Cauchy Convergence;275
9.13;13 Skorohod’s Representation Theorem;277
9.14;14 Problems;279
10;6 The Law of Large Numbers;284
10.1;1 Preliminaries;285
10.2;2 A Weak Law for Partial Maxima;288
10.3;3 The Weak Law of Large Numbers;289
10.4;4 A Weak Law Without Finite Mean;297
10.5;5 Convergence of Series;303
10.6;6 The Strong Law of Large Numbers;313
10.7;7 The Marcinkiewicz-Zygmund Strong Law;317
10.8;8 Randomly Indexed Sequences;320
10.9;9 Applications;324
10.10;10 Uniform Integrability; Moment Convergence;328
10.11;11 Complete Convergence;330
10.12;12 Some Additional Results and Remarks;334
10.13;13 Problems;342
11;7 The Central Limit Theorem;347
11.1;1 The i.i.d. Case;348
11.2;2 The Lindeberg-Levy-Feller Theorem;348
11.3;3 Anscombe’s Theorem;363
11.4;4 Applications;366
11.5;5 Uniform Integrability; Moment Convergence;370
11.6;6 Remainder Term Estimates;372
11.7;7 Some Additional Results and Remarks;380
11.8;8 Problems;394
12;8 The Law of the Iterated Logarithm;400
12.1;1 The Kolmogorov and Hartman-Wintner LILs;401
12.2;2 Exponential Bounds;402
12.3;3 Proof of the Hartman-Wintner Theorem;404
12.4;4 Proof of the Converse;413
12.5;5 The LIL for Subsequences;415
12.6;6 Cluster Sets;421
12.7;7 Some Additional Results and Remarks;429
12.8;8 Problems;437
13;9 Limit Theorems; Extensions and Generalizations;439
13.1;1 Stable Distributions;440
13.2;2 The Convergence to Types Theorem;443
13.3;3 Domains of Attraction;446
13.4;4 Infinitely Divisible Distributions;458
13.5;5 Sums of Dependent Random Variables;464
13.6;6 Convergence of Extremes;467
13.7;7 The Stein-Chen Method;475
13.8;8 Problems;480
14;10 Martingales;483
14.1;1 Conditional Expectation;484
14.2;2 Martingale Definitions;493
14.3;3 Examples;497
14.4;4 Orthogonality;503
14.5;5 Decompositions;505
14.6;6 Stopping Times;507
14.7;7 Doob’s Optional Sampling Theorem;511
14.8;8 Joining and Stopping Martingales;513
14.9;9 Martingale Inequalities;517
14.10;10 Convergence;524
14.11;11 The Martingale { E( Z | Fn)};531
14.12;12 Regular Martingales and Submartingales;532
14.13;13 The Kolmogorov Zero-one Law;536
14.14;14 Stopped Random Walks;537
14.15;15 Regularity;547
14.16;16 Reversed Martingales and Submartingales;557
14.17;17 Problems;564
15;A Some Useful Mathematics;570
15.1;1 Taylor Expansion;570
15.2;2 Mill’s Ratio;573
15.3;3 Sums and Integrals;574
15.4;4 Sums and Products;575
15.5;5 Convexity; Clarkson’s Inequality;576
15.6;6 Convergence of (Weighted) Averages;579
15.7;7 Regularly and Slowly Varying Functions;581
15.8;8 Cauchy’s Functional Equation;583
15.9;9 Functions and Dense Sets;585
16;References;591
17;Index;603