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