Gaussian Measures in Hilbert Space: Construction and Properties

Cover......Page 1
Half-Title Page......Page 3
Dedication......Page 4
Title Page......Page 5
Copyright Page......Page 6
Contents......Page 7
Foreword......Page 11
Preface......Page 15
 Acknowledgments......Page 16
Introduction......Page 17
Abbreviations and Notation......Page 21
 1.1. The change of variables formula......Page 25
 1.2. Invariance of Lebesgue measure......Page 28
 1.3. Absence of invariant measure in infinite-dimensional Hilbert space......Page 33
 1.4. Random vectors and their distributions......Page 34
  1.4.1. Random variables......Page 35
  1.4.2. Random vectors......Page 36
  1.4.3. Distributions of random vectors......Page 38
  1.5.1. Characteristic functions of Gaussian vectors......Page 41
  1.5.2. Expansion of Gaussian vector......Page 44
  1.5.3. Support of Gaussian vector......Page 46
  1.5.4. Gaussian measures in Euclidean space......Page 47
  2.1.1. Metric on R∞......Page 51
  2.1.2. Borel and cylindrical sigma-algebras coincide......Page 54
  2.1.3. Weighted l2 space......Page 55
  2.2.1. Kolmogorov extension theorem......Page 58
  2.2.2. Construction of product measure on B(R∞)......Page 60
  2.2.3. Properties of product measure......Page 62
 2.3. Standard Gaussian measure in R∞......Page 66
  2.3.1. Alternative proof of the second part of theorem 2.4......Page 69
 2.4. Construction of Gaussian measure in l2......Page 70
 3.1. Classes of operators in H......Page 75
  3.1.1. Hilbert–Schmidt operators......Page 76
  3.1.2. Polar decomposition......Page 79
  3.1.3. Nuclear operators......Page 81
  3.1.4. S-operators......Page 86
  3.2.1. Weak integral......Page 92
  3.2.2. Strong integral......Page 93
  3.3.1. Weak and strong moments......Page 99
  3.3.2. Examples of Borel measures......Page 102
  3.3.3. Boundedness of moment form......Page 107
 4.1. Cylindrical sigma-algebra in normed space......Page 113
 4.2. Convolution of measures......Page 117
 4.3. Properties of characteristic functionals in H......Page 120
 4.4. S-topology in H......Page 123
 4.5. Minlos–Sazonov theorem......Page 126
 5.1. Characteristic functional of Gaussian measure......Page 135
 5.2. Decomposition of Gaussian measure and Gaussian random element......Page 138
 5.3. Support of Gaussian measure and its invariance......Page 141
 5.4. Weak convergence of Gaussian measures......Page 149
  5.5.1. Gaussian measures in normed space......Page 153
  5.5.2. Fernique’s theorem......Page 157
 6.1. Uniformly integrable sequences......Page 167
  6.2.1. General properties of absolutely continuous measures......Page 169
  6.2.2. Kakutani’s theorem for product measures......Page 172
  6.2.3. Dichotomy for Gaussian product measures......Page 176
  6.3.1. The case where Gaussian measures have equal correlation operators......Page 179
  6.3.2. Necessary conditions for equivalence of Gaussian measures......Page 182
  6.3.3. Criterion for equivalence of Gaussian measures......Page 189
  6.4.1. Estimation and hypothesis testing for mean of Gaussian random element......Page 193
  6.4.2. Estimation and hypothesis testing for correlation operator of centered Gaussian random element......Page 197
 7.1. Solutions for Chapter 1......Page 203
 7.2. Solutions for Chapter 2......Page 217
  7.2.1. Generalized Kolmogorov extension theorem......Page 220
 7.3. Solutions for Chapter 3......Page 226
 7.4. Solutions for Chapter 4......Page 235
 7.5. Solutions for Chapter 5......Page 241
 7.6. Solutions for Chapter 6......Page 251
 Chapter 2......Page 259
 Chapter 4......Page 260
 Chapter 5......Page 261
 Chapter 6......Page 262
References......Page 263
Index......Page 265
Other titles from iSTE in Mathematics and Statistics......Page 269
EULA......Page 275