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PART FOUR:DEPENDENCE 3

CHAPTER Ⅷ:CONDITIONING 3

27．ONCEPT OF CONDITIONING 3

27.1 Elementary case 3

27.2 General case 7

27.3 Conditional expectation given a function 8

27.4 Relative conditional expectations and sufficient σ-fields 10

28．ROPERTIES OF CONDITIONING 13

28.1 Expectation properties 13

28.2 Smoothing properties 15

28.3 Concepts of conditional independence and of chains 17

29．REGULAR PR.FUNCTIONS 19

29.1 Regularity and integration 19

29.2 Decomposition of regular c.pr.'s given separable σ-fields 21

30．CONDITIONAL DISTRIBUTIONS 24

30.1 Definitions and restricted integration 24

30.2 Existence 26

30.3 Chains;the elementary case 31

COMPLEMENTS AND DETAILS 36

CHAPTER Ⅸ:FROM INDEPENDENCE TO DEPENDENCE 37

31．CENTRAL ASYMPTOTIC PROBLEM 37

31.1 Comparison of laws 38

31.2 Comparison ofsummands 41

31.3 Weighted prob.laws 44

32．CENTERINGS,MARTINGALES,AND A.S.CONVERGENCE 51

32.1 Centerings 51

32.3 Martingales:generalities 54

32.3 Martingales:convergence and closure 57

32.4 Applications 63

32.5 Indefinite expectations and a.s.convergence 67

COMPLEMENTS AND DETAILS 73

CHAPTER Ⅹ: ERGODIC THEOREMS 77

33．TRANSLATION OF SEQUENCES;BASIC ERGODIC THEOREM AND 77

STATIONARITY 77

33.1 Phenomenological origin 77

33.2 Basic ergodic inequality 79

33.3 Stationarity 83

33.4 Applications;ergodic hypothesis and independence 89

33.5 Applications;stationary chains 90

34．ERGODIC THEOREMS AND Lr-SPACES 96

34.1 Translations and their extensions 96

34.2 A.s.ergodic theorem 98

34.3 Ergodic theorems on spaces Lr 101

35．ERGODIC THEOREMS ON BANACH SPACES 106

35.1 Norms ergodic theorem 106

35.2 Uniform norms ergodic theorems 110

35.3 Application to constant chains 114

COMPLEMENTS AND DETAILS 118

CHAPTER Ⅺ:SECOND ORDER PROPERTIES 121

36．ORTHOGONALITY 121

36.1 Orthogonal r.v.'s; convergence and stability 122

36.2 Elementary orthogonal decomposition 125

36.3 Projection,conditioning,and normality 128

37．SECOND ORDER RANDOM FUNCTIONS 130

37.1 Covariances 131

37.2 Calculus in q.m.;continuity and differentiation 135

37.3 Calculus in q.m.;integration 137

37.4 Fourier-Stieltjes transforms in q.m 140

37.5 Orthogonal decompositions 143

37.6 Normality and almost-sure properties 151

37.7 A.s.stability 152

COMPLEMENTS AND DETAILS 156

PART FIVE: ELEMENTS OF RANDOM ANALYSIS 163

CHAPTER Ⅻ: FOUNDATIONS;MARTINGALES AND DECOMPOSABILITY 163

38．FOUNDATIONS 163

38.1 Generalities 164

38.2 Separability 170

38.3 Sample continuity 179

38.4 Random times 188

39．MARTINGALES 193

39.1 Closure and limits 194

39.2 Martingale times and stopping 207

40．DECOMPOSABILITY 212

40.1 Generalities 212

40.2 Three parts decomposition 216

40.3 Infinite decomposability;normal and Poisson cases 221

COMPLEMENTS AND DETAILS 231

CHAPTER ⅩⅢ: BROWNIAN MOTION AND LIMIT DISTRIBUTIONS 235

41．BROWNIAN MOTION 235

41.1 Origins 235

41.2 Definitions and relevant properties 237

41.3 Brownian sample oscillations 246

41.4 Brownian times and functionals 254

42．LIMIT DISTRIBUTIONS 263

42.1 Pr.'s on ？ 264

42.2 Limit distributions on ？ 268

42.3 Limit distributions;Brownian embedding 271

42.4 Some specific functionals 278

Complements and Details 281

CHAPTER ⅩⅣ: MARKOV PROCESSES 289

43．MARKOV DEPENDENCE 289

43.1 Markov property 289

43.2 Regular Markov processes 294

43.4 Stationarity 301

43.4 Strong Markov property 304

44．TIME-CONTINUOUS TRANSITION PROBABILITIES 310

44.1 Differentiation of tr.pr.'s 312

44.2 Sample functions behavior 321

45．MARKOV SEMI-GROUPS 331

45.1 Generalities 331

45.2 Analysis of semi-groups 336

45.3 Markov processes and semi-groups 346

46．SAMPLE CONTINUITY AND DIFFUSION OPERATORS 357

46.1 Strong Markov property and sample rightcontinuity 357

46.2 Extended infinitesimal operator 366

46.3 One-dimensional diffusion operator 374

COMPLEMENTS AND DETAILS 381

BIBLIOGRAPHY 384

INDEX 391