Gaussian Measures in Hilbert Space

Alexander Kukush is a Professor at Taras Shevchenko National University of Kyiv, Ukraine, where he teaches within its Faculty of Mechanics and Mathematics, and Department of Mathematical Analysis.

• Foreword ixPreface xiiiIntroduction xvAbbreviations and Notation xixChapter 1. Gaussian Measures in Euclidean Space 11.1. The change of variables formula 11.2. Invariance of Lebesgue measure 41.3. Absence of invariant measure in infinite-dimensional Hilbert space 91.4. Random vectors and their distributions 101.4.1. Random variables 111.4.2. Random vectors 121.4.3. Distributions of random vectors 141.5. Gaussian vectors and Gaussian measures 171.5.1. Characteristic functions of Gaussian vectors 171.5.2. Expansion of Gaussian vector 201.5.3. Support of Gaussian vector 221.5.4. Gaussian measures in Euclidean space 23Chapter 2. Gaussian Measure in l2 as a Product Measure 272.1. Space R∞ 272.1.1. Metric on R∞ 272.1.2. Borel and cylindrical sigma-algebras coincide 302.1.3. Weighted l2 space 312.2. Product measure in R∞ 342.2.1. Kolmogorov extension theorem 342.2.2. Construction of product measure on B(R∞) 362.2.3. Properties of product measure 382.3. Standard Gaussian measure in R∞ 422.3.1. Alternative proof of the second part of theorem 2.4 452.4. Construction of Gaussian measure in l2 46Chapter 3. Borel Measures in Hilbert Space 513.1. Classes of operators in H 513.1.1. Hilbert–Schmidt operators 523.1.2. Polar decomposition 553.1.3. Nuclear operators 573.1.4. S-operators 623.2. Pettis and Bochner integrals 683.2.1. Weak integral 683.2.2. Strong integral 693.3. Borel measures in Hilbert space 753.3.1. Weak and strong moments 753.3.2. Examples of Borel measures 783.3.3. Boundedness of moment form 83Chapter 4. Construction of Measure by its Characteristic Functional 894.1. Cylindrical sigma-algebra in normed space 894.2. Convolution of measures 934.3. Properties of characteristic functionals in H 964.4. S-topology in H 994.5. Minlos–Sazonov theorem 102Chapter 5. Gaussian Measure of General Form 1115.1. Characteristic functional of Gaussian measure 1115.2. Decomposition of Gaussian measure and Gaussian random element 1145.3. Support of Gaussian measure and its invariance 1175.4. Weak convergence of Gaussian measures 1255.5. Exponential moments of Gaussian measure in normed space 1295.5.1. Gaussian measures in normed space 1295.5.2. Fernique’s theorem 133Chapter 6. Equivalence and Singularity of Gaussian Measures 1436.1. Uniformly integrable sequences 1436.2. Kakutani’s dichotomy for product measures on R∞ 1456.2.1. General properties of absolutely continuous measures 1456.2.2. Kakutani’s theorem for product measures 1486.2.3. Dichotomy for Gaussian product measures 1526.3. Feldman–Hájek dichotomy for Gaussian measures on H 1556.3.1. The case where Gaussian measures have equal correlation operators 1556.3.2. Necessary conditions for equivalence of Gaussian measures 1586.3.3. Criterion for equivalence of Gaussian measures 1656.4. Applications in statistics 1696.4.1. Estimation and hypothesis testing for mean of Gaussian random element 1696.4.2. Estimation and hypothesis testing for correlation operator of centered Gaussian random element 173Chapter 7. Solutions 1797.1. Solutions for Chapter 1 1797.2. Solutions for Chapter 2 1937.2.1. Generalized Kolmogorov extension theorem 1967.3. Solutions for Chapter 3 2027.4. Solutions for Chapter 4 2117.5. Solutions for Chapter 5 2177.6. Solutions for Chapter 6 227Summarizing Remarks 235References 239Index 241