PART FOUR: DEPENDENCE CHAPTER ��: 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 ��-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 ��: 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 ��: 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 �� 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 ��: 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 ����: 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 ���� 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 operator COMPLEMENTS AND DETAILS BIBLIOGRAPHY INDEX