Probability Distributions on Banach Spaces

I. Measurability and Measures.- 1. ?-algebras and measurable mappings in metric space.- 1. Measurable spaces and mappings, Borel ?-algebra (1). 2. ?-algebras generated by families of mappings (3). 3. ?-algebras in product spaces (8). 4. The structure of measurable mappings (10). Exercises (14)..- 2. ?-algebras in Banach spaces.- 1. Measurable vector spaces (15)..- 2. Relations between various ?-algebras (16)..- 3. Auxiliary results and counter-examples (20). Exercises (24)..- 3. Probability measures on topological spaces.- 1. Preliminaries from the general measure theory (25). 2. Regular, ?-smooth and Radon measures (27). 3. The support of a measure (30). 4. Existence of ?-smooth and Radon extensions (33). 5. Extensions of Baire measures. Prokhorov’s theorem (38). 6. Embedding of the space of measures into the space of linear functionals (40). 7. Weak convergence of probability measures (41). 8. The weak topology. Prokhorov’s criterion (47). 9. Criteria of weak relative compactness of families of probability measures on a Banach space (51). Exercises (54)..- 4. The semigroup of probability measures.- 1. Extensions of products of measures (58)..- 2. Definition of the convolution of probability measures (63). 3. Continuity of convolution. Equation ?*v = ? (65). 4. Weak compactness and convolution (68). Exericses (71)..- 5. Invariant and quasi-invariant measures. Scalarly non-degenerate measures.- 1. Existence of invariant measures, A. Weil’s theorem and its consequences (72). 2. Admissible translations and quasi-invariance (74). 3. Scalarly non-degenerate measures. Extension of weakly ?-smooth and weakly Radon measures (76). Exercises (81).- Supplementary comments.- II. Measures and Random Elements. Weak and Strong Orders.- 1. Random elements.- 1. Probability spaces (87). 2. Random elements (88). 3. Types of convergence of random elements. Space L0(?,A,P;X) (91). 4. Structure of operators generated by random elements (95). Exercises (98)..- 2. Weak and strong orders of random elements.- 1. Comparison of various classes (102)..- 2. The strong order. Connections with summing and nuclear operators (104). 3. The weak order (108). 4. The case of Hilbert space. Connections with Hilbert-Schmidt operators (109). Exercises (111)..- 3. The expectation.- 1. Definition and existence (113)..- 2. Indefinite integral. The Radon-Nikodym Property (120). Exercises (123)..- 4. Conditional expectations, martingales and connections with the Radon-Nikodym Property.- 1. Conditional expectations (125). 2. Convergence in the mean of martingales (128)..- 3. A.s. boundedness and the a.s. convergence of martingales (130). 4. Convergence of martingales and the Radon-Nikodym property (135). Exercises (138)..- Supplementary comments.- III. Covariance Operators.- 1. Operators mapping spaces into their duals.- 1. General properties (144). 2. A factorization lemma (148). 3. Connection with the reproducing kernel Hilbert space (154). 4. Symmetric compact operators on a Hilbert space (159). 5. Positive-definite operator functions (162). 6. The case of complex spaces (165). Exercises (167)..- 2. Covariance operators.- 1. Definition and properties (168)..- 2. Description of the class of covariance operators (173). 3. Covariance operators of probability measures of strong order two (174). 4. Operators of cross-correlation (177). 5. Remarks about the case of complex
spaces (180). Exercises (181)..- Supplementary comments.- IV. Characteristic Functionals.- 1. Positive-definite functions.- 1. Definition and fundamental properties (184). 2. Positive-definite functions on groups (188). 3. Examples of positive-definite functions (189). 4. Negative-definite functions (190). 5. Integrals of positive-definite functions (194). Exercises (195)..- 2. Definition and general properties of characteristic functionals.- 1. Definition and uniqueness (197). 2. Other properties of characteristic functionals (202). 3. Continuity (206). 4. An important example: Gaussian measures (209). 5. Characteristic functionals induced by operators (216). 6. An application to the problem of
characterization of random elements with Radon distributions (218). 7. Remarks on the case of complex spaces (220). Exercises (221)..- 3. Characteristic functionals and weak convergence.- 1. Conditions for weak convergence (223)..- 2. Levy’s theorem (225). 3. Weak convergence in the weak topology (229). 4. Continuity of operators with values in L0 (231). Exercises (232)..- 4. Bochner’s theorem.- 1. Bochner’s theorem in Rn (233). 2. An infinite-dimensional analogue of Bochner’s theorem: Bochner-Kolmogorov’s theorem (235).
3. Isometrically invariant measures. Schoenberg’s theorem (236). Exercises (241)..- Supplementary comments.- V. Sums of Independent Random Elements.- 1. Independent random elements.- 1. Basic definitions (250). 2. The zero-one law (253). Exercises (258)..- 2. Series of independent random elements.- 1. Sums of independent random elements and convolution (260). 2. Levy’s inequality and its consequences (261). 3. Convergence and boundedness of series of sign-invariant sequences of random elements (265). 4. Equivalence of various types of convergence (268). 5. The case of independent symmetric summands (271). Exercises (276)..- 3. Integrability of sums and the mean convergence of random series.- 1. Auxiliary inequalities (278). 2. Exponential integrability of sums of series of uniformly bounded summands (285). 3. Mean convergence of series. The general case (290). Exercises (296)..- 4. Comparison of random series.- 1. Bounded multipliers (298). 2. Connections with the unconditional convergence (302). 3. Unconditional convergence in L0(?,X) (306). 4. Comparison of series of the form ?xkfk and ?xkgk (310). Exercises (314)..- 5. Some special series.- 1. General series of the form ?xn fn (316).
2. Series of the form ?xn?n (320). 3. The case of Gaussian series (326). 4. Convergence of Gaussian series and a description of Gaussian covariances (332). 5. Expansion of Gaussian random elements (335). 6. Series ?xn fn for stable fn (338).
Exercises (344)..- 6. Random series in spaces which do not contain c0.- 1. Boundedness and convergence (347)..- 2. Uniform convergence (353). Exercises (356)..- Supplementary comments.- VI. Topological Description of Characteristic Functionals and Cylindrical Measures.- 1. Sazonov’s theorem and related topics.- 1. Sazonov’s theorem (362). 2. Compactness of families of probability measures and equicontinuity of characteristic functionals (365). 3. Characteristic functionals of measures with Hilbertian support (368). 4. The case of dual spaces (372). Exercises (373)..- 2. Necessary and sufficient topologies. Spaces with the Sazonov property.- 1. Definitions and fundamental theorem (374).
2. Embedding in L0 and the Sazonov property (380). 3. Admissible topologies different from ?nuc (H) (385). Exercises (388)..- 3. Cylindrical measures.- 1. Properties of cylindrical measures (390).
2. Cylindrical measures and operators acting into L0 (395). 3. Prokhorov’s theorem and extensions of cylindrical measures (396). 4. Convolution of cylindrical measures (398). 5. Measures with weakly continuous characteristic functionals (401). Exercises (403)..- 4. The case of locally convex spaces. Minlos’ theorem.- 1. Auxiliary lemmas (406). 2. The sufficiency of topology $$\tau _{{\rm \overline S}} {\rm (X*,X)}$$ (408).
3. The case of nuclear spaces, Minlos’ theorem (409). Exercises (411)..- 5. Radonifying operators.- 1. Continuity and type of a cylindrical measure (412). 2. (q,p)-radonifying operators (416). 3. P-summing and p-Radonifying operators (l