概率论（第四版）

INTRODUCTORY PART: ELEMENTARY PROBABILITY THEORY
SECTION
I. INTUITIVE BACKGROUND
1. Events
2. Random events and trials
3. Random variables
II. AXIOMS; INDEPENDENCE AND THE BERNOULLI CASE
1. Axioms of the finite case
2. Simple random variables
3. Independence
4. Bernoulli case
5. Axioms for the countable case
6. Elementary random variables
7. Need for nonelementary random variables
III. DEPENDENCE AND CHAINS
1. Conditional probabilities
2. Asymptotically Bernoullian case
3. Recurrence
4. Chain dependence
5. Types of states and asymptotic behavior
6. Motion of the system
7. Stationary chains
COMPLEMENTS AND DETAILS
PART ONE: NOTIONS OF MEASURE THEORY
CHAPTER I: SETS, SPACES, AND MEASURES
1. SETS, CLASSES, AND FUNCTIONS
1.1 Definitions and notations
1.2 Differences, unions, and intersections
1.3 Sequences and limits
1.4 Indicators of sets
1.5 Fields and -fields
1.6 Monotone classes
1.7 Product sets
1.8 Functions and inverse functions
1.9 Measurable spaces and functions
2. TOPOLOGICAL SPACES
2.1 Topologies and limits
2.2 Limit points and compact spaces
2.3 Countability and metric spaces
2.4 Linearity and normed spaces
3. ADDITIVE SET FUNCTIONS
3.1 Additivity and continuity
3.2 Decomposition of additive set functions
4. CONSTRUCTION OF MEASURES ON -FIELDS
4.1 Extension of measures
4.2 Product probabilities
4.3 Consistent probabilities on Borel fields
4.4 Lebesgue-Stieltjes measures and distribution functions
COMPLEMENTS AND DETAILS
CHAPTER II: MEASURABLE FUNCTIONS AND INTEGRATION
5. MEASURABLE FUNCTIONS
5.1 Numbers
5.2 Numerical functions
5.3 Measurable functions
6. MEASURE AND CONVERGENCES
6.1 Definitions and general properties
6.2 Convergence almost everywhere
6.3 Convergence in measure
7. INTEGRATION
7.1 Integrals
7.2 Convergence theorems
8. INDEFINITE INTEGRALS; ITERATED INTEGRALS
8.1 Indefinite integrals and Lebesgue decomposition
8.2 Product measures and iterated integrals
8.3 Iterated integrals and infinite product spaces
COMPLEMENTS AND DETAILS
PART TWO: GENERAL CONCEPTS AND TOOLS OF
PROBABILITY THEORY
CHAPTER III: PROBABILITY CONCEPTS
9. PROBABILITY SPACES AND RANDOM VARIABLES
9.1 Probability terminology
*9.2 Random vectors, sequences, and functions
9.3 Moments, inequalities, and convergences
*9.4 Spaces Lr
10. PROBABILITY DISTRIBUTIONS
10.1 Distributions and distribution functions
10.2 The essential feature of pr. theory
COMPLEMENTS AND DETAILS
CHAPTER IV: DISTRIBUTION FUNCTIONS AND CHARACTERISTIC FUNCTIONS
11. DISTRIBUTION FUNCTIONS
11.1 Decomposition
11.2 Convergence of d.f.''s
11.3 Convergence of sequences of integrals
*11.4 Further extension and convergence of moments
11.5 Discussion
12. CONVERGENCE OF PROBABILITIES ON METRIC SPACES
12.1 Convergence
12.2 Regularity and tightness
12.3 Tightness and relative compactness
13. CHARACTERISTIC FUNCTIONS AND DISTRIBUTION FUNCTIONS
13.1 Uniqueness
13.2 Convergences
13.3 Composition of d.f.''s and multiplication of ch.f.''s
13.4 Elementary properties of ch.f.''s and first applications
14. PROBABILITY LAWS AND TYPES OF LAWS
14.1 Laws and types; the degenerate type
14.2 Convergence of types
14.3 Extensions
15. NONNEGATIVE-DEFINITENESS; REGULARITY
15.1 Ch.f.''s and nonnegative-definiteness
15.2 Regularity and extension of ch.f.''s
15.3 Composition and decomposition of regular ch.f.''s
COMPLEMENTS AND DETAILS
PART THREE: INDEPENDENCE
CHAPTER V: SUMS OF INDEPENDENT RANDOM VARIABLES
16. CONCEPT OF INDEPENDENCE
16.1 Independent classes and independent functions
16.2 Multiplication properties
16.3 Sequences of independent r.v.''s
16.4 Independent r.v.''s and product spaces
17. CONVERGENCE AND STABILITY OF SUMS; CENTERING AT
EXPECTATIONS AND TRUNCATION
17.1 Centering at expectations and truncation
17.2 Bounds in terms of variances
17.3 Convergence and stability
17.4 Generalization
18. CONVERGENCE AND STABILITY OF SUMS; CENTERING AT
MEDIANS AND SYMMETRIZATION
18.1 Centering at medians and symmetrization
18.2 Convergence and stability
19. EXPONENTIAL BOUNDS AND NORMED SUMS
19.1 Exponential bounds
19.2 Stability
19.3 Law of the iterated logarithm
COMPLEMENTS AND DETAILS
CHAPTER VI: CENTRAL LIMIT PROBLEM
20. DEGENERATE, NORMAL, AND POISSON TYPES
20.1 First limit theorems and limit laws
20.2 Composition and decomposition
21. EVOLUTION OF THE PROBLEM
21.1 The problem and preliminary solutions
21.2 Solution of the Classical Limit Problem
21.3 Normal approximation
22. CENTRAL LIMIT PROBLEM; THE CASE OF BOUNDED VARIANCES
22.1 Evolution of the problem
22.2 The case of bounded variances
23. SOLUTION OF THE CENTRAL LIMIT PROBLEM
23.1 A family of limit laws; the infinitely decomposable laws
23.2 The uan condition
23.3 Central Limit Theorem
23.4 Central convergence criterion
23.5 Normal, Poisson, and degenerate convergence
24. NORMED SUMS
24.1 The problem
24.2 Norming sequences
24.3 Characterization of
24.4 Identically distributed summands and stable laws
24.5 Levy representation
COMPLEMENTS AND DETAILS
CHAPTER VII: INDEPENDENT IDENTICALLY DISTRIBUTED SUMMANDS
25. REGULAR VARIATION AND DOMAINS OF ATTRACTION
25.1 Regular variation
25.2 Domains of attraction
26. RANDOM WALK
26.1 Set-up and basic implications
26.2 Dichotomy: recurrence and transience
26.3 Fluctuations; exponential identities
26.4 Fluctuations; asymptotic behaviour
COMPLEMENTS AND DETAILS
BIBLIOGRAPHY
INDEX