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Overview
"This is a very scholarly book in the best tradition of analysis. Nothing else of this type exists for the benefit of the serious student of the subject and it is safe to predict that it will remain a standard compendium for many years to come." ― S. Vajda in Zentralblatt für Mathematik
In the decades following its 1963 publication, this volume served as the standard advanced text in probability theory. Geared toward graduate students and professionals in the field of probability and statistics, the treatment offers extensive introductory material and is suitable for an undergraduate course in probability theory. The first four chapters cover notions of measure theory plus general concepts and tools of probability theory. Subsequent chapters explore sums of independent random variables, the central limit problem, conditioning, independence and dependence, ergodic theorems, and second order properties. The final two chapters examine foundations, martingales, and decomposability as well as Markov processes.
Product Details
| ISBN-13: | 9780486814889 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 07/18/2017 |
| Series: | Dover Books on Mathematics |
| Pages: | 704 |
| Sales rank: | 1,140,075 |
| Product dimensions: | 6.00(w) x 8.80(h) x 1.40(d) |
About the Author
Table of Contents
Introductory Part: Elementary Probability Theory
I Intuitive Background 3
1 Events 3
2 Random events and trials 5
3 Random variables 6
II Axioms; Independence and the Bernoulli Case 8
1 Axioms of the finite case 8
2 Simple random variables 9
3 Independence 11
4 Bernoulli case 12
5 Axioms for the countable case 15
6 Elementary random variables 17
7 Need for nonelementary random variables 22
III Dependence and Chains 24
1 Conditional probabilities 24
2 Asymptotically Bernoullian case 25
3 Recurrence 26
4 Chain dependence 28
*5 Types of states and asymptotic behavior 30
*6 Motion of the system 36
*7 Stationary chains 39
Complements and Details 42
Part 1 Notions of Measure Theory
Chapter I Sets, Spaces, and Measures
1 Sets, Classes, and Functions 55
1.1 Definitions and notations 55
1.2 Differences, unions, and intersections 56
1.3 Sequences and limits 57
1.4 Indicators of sets 59
1.5 Fields and τ-fields 59
1.6 Monotone classes 60
*1.7 Product sets 61
*1.8 Functions and inverse functions 62
*1.9 Measurable spaces and functions 64
*2 Topological Spaces 65
*2.1 Topologies and limits 66
*2.2 Limit points and compact spaces 69
*2.3 Countability and metric spaces 72
*2.4 Linearity and normed spaces 77
3 Additive Set Functions 82
3.1 Additivity and continuity 82
3.2 Decomposition of additive set functions 86
*4 Construction of Measures on τ-Fields 87
*4.1 Extension of measures 87
*4.2 Product probabilities 90
*4.3 Consistent probabilities on Borel fields 92
*4.4 Lebesgue-Stieltjes measures and distribution functions 95
Complements and Details 99
Chapter II Measurable Functions and Integration
5 Measurable Functions 102
5.1 Numbers 102
5.2 Numerical functions 104
5.3 Measurable functions 106
6 Measure and Convergences 110
6.1 Definitions and general properties 110
6.2 Convergence almost everywhere 113
6.3 Convergence in measure 115
7 Integration 117
7.1 Integrals 118
7.2 Convergence theorems 124
8 Indefinite Integrals; Iterated Integrals 129
8.1 Indefinite integrals and Lebesgue decomposition 129
8.2 Product measures and iterated integrals 134
*8.3 Iterated integrals and infinite product spaces 136
Complements and Details 138
Part 2 General Concepts and Tools of Probability Theory
Chapter III Probability Concepts
9 Probability Spaces and Random Variables 149
9.1 Probability terminology 149
*9.2 Random vectors, sequences, and functions 153
9.3 Moments, inequalities, and convergences 154
*9.4 Spaces Lr 160
10 Probability Distributions 166
10.1 Distributions and distribution functions 166
10.2 The essential feature of pr. theory 170
Complements and Details 172
Chapter IV Distribution Functions and Characteristic Functions
11 Distribution Functions 175
11.1 Decomposition 175
11.2 Convergence of d.f.'s 178
11.3 Convergence of sequences of integrals 180
*11.4 Final extension and convergence of moments 182
12 Characteristic Functions and Distribution Functions 185
12.1 Uniqueness 186
12.2 Convergences 189
12.3 Composition of d.f.'s and multiplication of ch.f.'s 193
12.4 Elementary properties of ch.f.'s and first applications 194
13 Probability Laws and Types of Laws 201
13.1 Laws and types; the degenerate type 201
13.2 Convergence of types 203
13.3 Extensions 205
14 Nonnegative-Definiteness; Regularity 205
14.1 Ch.f.'s and nonnegative-definiteness 205
*14.2 Regularity and extension of ch.f.'s 210
*14.3 Composition and decomposition of regular ch.f.'s 213
Complements and Details 214
Part 3 Independence
Chapter V Sums of Independent Random Variables
15 Concept of Independence 223
15.1 Independent classes and independent functions 223
15.2 Multiplication properties 226
15.3 Sequences of independent r.v.'s 228
*15.4 Independent r.v.'s and product spaces 230
16 Convergence and Stability of Sums; Centering at Expectations and Truncation 231
16.1 Centering at expectations and truncation 232
16.2 Bounds in terms of variances 234
16.3 Convergence and stability 236
*16.4 Generalization 240
*17 Convergence and Stability of Sums; Centering at Medians and Symmetrization 243
*17.1 Centering at medians and symmetrization 244
*17.2 Convergence and stability 248
*18 Exponential Bounds and Normed Sums 254
*18.1 Exponential bounds 254
*18.2 Stability 258
*18.3 Law of the iterated logarithm 260
Complements and Details 263
Chapter VI Central Limit Problem
19 Degenerate, Normal, and Poisson Types 268
19.1 First limit theorems and limit laws 268
*19.2 Composition and decomposition 271
20 Evolution of the Problem 274
20.1 The problem and preliminary solutions 274
20.2 Solution of the Classical Limit Problem 278
*20.3 Normal approximation 282
21 Central Limit Problem; The Case of Bounded Variances 288
21.1 Evolution of the problem 288
21.2 The case of bounded variances 290
*22 Solution of the Central Limit Problem 296
*22.1 A family of limit laws; the infinitely decomposable laws 296
*22.2 The uan condition 302
*22.3 Central Limit Theorem 307
*22.4 Central convergence criterion 311
*22.5 Normal, Poisson, and degenerate convergence 315
*23 Normed Sums 319
*23.1 The problem 319
*23.2 Normisig sequences 320
*23.3 Characterization of π 322
*23.4 Identically distributed summands and stable laws 326
Complements and Details 331
Part 4 Dependence
Chapter VII Conditioning
24 Concept of Conditioning 337
24.1 Elementary case 337
24.2 General case 341
24.3 Conditional expectation given a function 342
*24.4 Relative conditional expectations and sufficient σ-fields 344
25 Properties of Conditioning 347
25.1 Expectation properties 347
25.2 Smoothing properties 349
*25.3 Concepts of conditional independence and of chains 351
26 Regular Pr. Functions 353
26.1 Regularity and integration 353
*26.2 Decomposition of regular c.pr.'s given separable τ-fields 355
27 Conditional Distributions 358
27.1 Definitions and restricted integration 358
27.2 Existence 360
27.3 Chains; the elementary case 365
Complements and Details 370
Chapter VIII From Independence to Dependence
28 Central Asymptotic Problem 371
28.1 Comparison of laws 372
28.2 Comparison of summands 375
*28.3 Weighted limit laws 378
29 Centerings, Martingales, and A.S. Convergence 385
29.1 Centerings 385
29.2 Martingales: generalities 388
29.3 Martingales: convergence and closure 391
29.4 Applications 397
*29.5 Indefinite expectations and a.s. convergence 401
Complements and Details 407
Chapter IX Ergodic Theorems
30 Translation of Sequences; Basic Ergodec Theorem and Stationarity 411
*30.1 Phenomenological origin 411
30.2 Basic ergodic inequality 413
30.3 Stationarity 417
30.4 Applications; ergodic hypothesis and independence 423
*30.5 Applications; stationary chains 424
*31 Ergodic Theorems and Lr-Spaces 430
*31.1 Translations and their extensions 430
*31.2 A.s. ergodic theorem 432
*31.3 Ergodic theorems on spaces Lr 435
*32 Ergodic Theorems on Banach Spaces 440
*32.1 Norms ergodic theorem 440
*32.2 Uniform norms ergodic theorems 444
*32.3 Application to constant chains 448
Complements and Details 452
Chapter X Second Order Properties
33 Orthogonality 455
33.1 Orthogonal r.v.'s; convergence and stability 456
33.2 Elementary orthogonal decomposition 459
33.3 Projection, conditioning, and normality 462
34 Second Order Random Functions 464
34.1 Covariances 465
34.2 Calculus in q.m.; continuity and differentiation 469
34.3 Calculus in q.m.; integration 471
34.4 Fourier-Stieltjes transforms in q.m 474
34.5 Orthogonal decompositions 477
34.6 Normality and almost-sure properties 485
34.7 A.s. stability 486
Complements and Details 490
Part 5 Elements of Random Analysis
Chapter XI Foundations; Martingales and Decomposability
35 Foundations 497
35.1 Generalities 498
35.2 Separability 504
35.3 Sample continuity 513
36 Martingales 522
36.1 Continuity 522
36.2 Martingale times 530
37 Decomposability 535
37.1 Generalities 536
37.2 Three parts decomposition 540
37.3 Infinite decomposability; normal and Poisson cases 545
Complements and Details 554
Chapter XII Markov Processes
38 Markov Dependence 562
38.1 Markov property 562
38.2 Regular Markov processes 567
38.3 Stationarity 574
38.4 Strong Markov property 577
39 Time-Continuous Transition Probabilities 583
39.1 Differentiation of tr. pr.'s 585
39.2 Sample functions behavior 594
40 Markov Semi-Groups 604
40.1 Generalities 604
40.2 Analysis of semi-groups 609
40.3 Markov processes and semi-groups 619
41 Sample Continuity and Diffusion Operators 630
41.1 Strong Markov property and sample rightcontinuity 630
41.2 Extended infinitesimal operator 639
41.3 One-dimensional diffusion operator 647
Complements and Details 654
Bibliography 657
Index 665