PART FOUR: DEPENDENCE
CHAPTER Ⅷ: CONDITIONING
27. CONCEPT OF CONDITIONING
27.1 Elementary case
27.2 General case
27.3 Conditional expectation given a function
*27.4 Relative conditional expectations and sufficient
σ-fiields
28. PROPERTIES OF CONDITIONING
28.1 Expectation properties
28.2 Smoothing properties
*28.3 Concepts of conditional independence and of chains
29. REGULAR PR. FUNCTIONS
29.1 Regularity and integration
*29.2 Decomposition of regular c.pr.'s given separable
a-fields
30. CONDITIONAL DISTRIBUTIONS
30.1 Definitions and restricted integration
30.2 Existence.
30.3 Chains; the elementary case
COMPLEMENTS AND DETAILS
CHAPTER Ⅸ: FROM INDEPENDENCE TO DEPENDENCE
31. CENTRAL ASYMPTOTIC PROBLEM
31.1 Comparison of laws
31.2 Comparison of summands
"31.3 Weighted prob. laws
32. CENTERINGS, MARTINGALES, AND A.$. CONVERGENCE
32.1 Centerings
32.3 Martingales: generalities
32.3 Martingales: convergence and closure
32.4 Applications
*32.5 Indefinite expectations and a.s. convergence
COMPLEMENTS AND DETAILS
CHAPTER Ⅹ: ERGODIC THEOREMS
33. TRANSLATION OF SEQUENCES; BASIC ERGODIC THEOREM AN
STATIONA RITY
*33.1 Phenomenological origin
33.2 Basic ergodic inequality
33.3 Stationarity
33.4 Applications; ergodic hypothesis and independence
*33.5 Applications; stationary chains
*34. ERGODIC THEOREMS AND Lt-SPACES
*34.1 Translations and their extensions
*34.2 A.s. ergodic theorem
*34.3 Ergodic theorems on spaces L
*35. ERGODIC THEOREMS ON BANACH SPACES
*35.1 Norms crgodic theorem
*35.2 Uniform norms ergodic theorems
*35.3 Application to constant chains
COMPLEMENTS AND DETAILS
CHAPTER Ⅺ SECOND ORDER PROPERTIES
36. ORTHOGONALITY
36.1 Orthogonal r.v.'s; convergence and stability
36.2 Elementary orthogonal decomposition
36.3 Projection, conditioning, and normality
37. SECOND ORDER RANDOM FUNCTIONS
37.1 Covarianccs
37.2 Calculus in q.m.; continuity and differentiation
37.3 Calculus in q.m.; integration
37.4 Fourier-Stichjes transforms in q.m.
37.5 Orthogonal decompositions
37.6 Normality and almost-sure properties
37.7 A.s. stability
COMPLEMENTS AND DETAILS
PART FIVE: ELEMENTS OF RANDOM ANALYSIS
CHAPTER Ⅻ: FOUNDATIONS; MARTINGALES AND DECOMPOSABILITY
38. FOUNDATIONS
38.1 Generalities
38.2 Separability
38.3 Sample continuity
38.4 Random times
39. MARTINGALES .
39.1 Closure and limits
39.2 Martingale times and stopping
40. DECOMPOSABILITY
40.1 Generalities
40.2 Three parts decomposition
40.3 Infinite decomposability; normal and Poisson cases
COMPLEMENTS AND DETAILS
CHAPTER ⅩⅢ: BROWNIAN MOTION AND LIMIT DISTRIBUTIONS
41. BROWNIAN MOTION
41.1 Origins
41.2 Definitions and relevant properties
41.3 Brownian sample oscillations
41.4 Brownian times and functionals
42. LIMIT DISTRIBUTIONS
42.1 Pr.'son
42.2 Limit distributions on e.
42.3 Limit distributions; Brownian embedding
42.4 Some specific functionals
Complements and Details
CHAPTER ⅩⅣ MARKOV PROCESSES
43. MARKOV DEPENDENCE
43.1 Markov property
43.2 Regular Markov processes
43.4 Stationarity
43.4 Strong Markov property
44. TIME-CONTINUOUS TRANSITION PROBABILITIES
44.1 Differentiation of tr. pr.'s
44.2 Sample functions behavior
45. MARKOV SEMI-GROUPS
45.1 Generalities
45.2 Analysis of semi-groups
45.3 Markov processes and semi-groups
46. SAMPLE CONTINUITY AND DIFFUSION OPERATORS
46.1 Strong Markov property and sample rightcontinuity
46.2 Extended infinitesimal operator
46.3 One-dimensional diffusion o

本书被公认为是一套概率论方面的标准经典教科书，供高年级大学生和研究生使用，同时也是概率论和统计学方面研究人员经常使用的参考书。本书把概率论建立在严格的逻辑基础上，理论体系完整。第2卷包括两部分内容，涉及条件运算及独立随机变量和极限性质的相依性、二阶随机函数、随机分析的基本概念以及鞅、可分解性、随机函数的马尔可夫型等。读者对象：数学及相关专业的研究生。