Probability Distributions on Banach Spaces - N. Vakhania

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Sommaire:
I. Measurability and Measures.- 1. ?-algebras and measurable mappings in metric space.- 2. ?-algebras in Banach spaces.- 3. Probability measures on topological spaces.- 4. The semigroup of probability measures.- 5. Invariant and quasi-invariant measures. Scalarly non-degenerate measures.- Supplementary comments.- II. Measures and Random Elements. Weak and Strong Orders.- 1. Random elements.- 2. Weak and strong orders of random elements.- 3. The expectation.- 4. Conditional expectations, martingales and connections with the Radon-Nikodym Property.- Supplementary comments.- III. Covariance Operators.- 1. Operators mapping spaces into their duals.- 2. Covariance operators.- Supplementary comments.- IV. Characteristic Functionals.- 1. Positive-definite functions.- 2. Definition and general properties of characteristic functionals.- 3. Characteristic functionals and weak convergence.- 4. Bochner's theorem.- Supplementary comments.- V. Sums of Independent Random Elements.- 1. Independent random elements.- 2. Series of independent random elements.- 3. Integrability of sums and the mean convergence of random series.- 4. Comparison of random series.- 5. Some special series.- 6. Random series in spaces which do not contain c0.- Supplementary comments.- VI. Topological Description of Characteristic Functionals and Cylindrical Measures.- 1. Sazonov's theorem and related topics.- 2. Necessary and sufficient topologies. Spaces with the Sazonov property.- 3. Cylindrical measures.- 4. The case of locally convex spaces. Minlos' theorem.- 5. Radonifying operators.- Supplementary comments.