PART FOUR: DEPENDENCE
CHAPTER VIII: CONDITIONING
SECTION
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
a-fields
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 surnmands
31.3 Weighted prob. laws
32. CENTERINGS, MARTINGALES, AND A.S. CONVERGENCE
32.1 Centerings
32.3 Martingales: generalities
SECTION
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 AND
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 Lr-SPACES
*34.1 Translations and their extensions
*34.2 A.s. ergodic theorem
*34.3 Ergodic theorems on spaces Lr
*35. ERGODIC THEOREMS ON BANACH SPACES
*35.1 Norms ergodic 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 Covariances
37.2 Calculus in q.m.; continuity and differentiation
37.3 Calculus in q.m.; integration
37.4 Fourier-Stieltjes 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
SECTION
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
CHAPTEK 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.''s on e
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
SECTION
44. TIME-CONTINUOUS TRANSlTION 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 COSTINUITY 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
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