Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction: Donsker's Theorem, Metric Entropy, and Inequalities
- 2 Gaussian Measures and Processes; Sample Continuity
- 3 Foundations of Uniform Central Limit Theorems: Donsker Classes
- 4 Vapnik-Červonenkis Combinatorics
- 5 Measurability
- 6 Limit Theorems for Vapnik-Červonenkis and Related Classes
- 7 Metric Entropy, with Inclusion and Bracketing
- 8 Approximation of Functions and Sets
- 9 Sums in General Banach Spaces and Invariance Principles
- 10 Universal and Uniform Central Limit Theorems
- 11 The Two-Sample Case, the Bootstrap, and Confidence Sets
- 12 Classes of Sets or Functions Too Large for Central Limit Theorems
- Appendix A Differentiating under an Integral Sign
- Appendix B Multinomial Distributions
- Appendix C Measures on Nonseparable Metric Spaces
- Appendix D An Extension of Lusin's Theorem
- Appendix E Bochner and Pettis Integrals
- Appendix F Nonexistence of Types of Linear Forms on Some Spaces
- Appendix G Separation of Analytic Sets; Borel Injections
- Appendix H Young-Orlicz Spaces
- Appendix I Modifications and Versions of Isonormal Processes
- Subject Index
- Author Index
- Index of Notation
Appendix I - Modifications and Versions of Isonormal Processes
Published online by Cambridge University Press: 20 May 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction: Donsker's Theorem, Metric Entropy, and Inequalities
- 2 Gaussian Measures and Processes; Sample Continuity
- 3 Foundations of Uniform Central Limit Theorems: Donsker Classes
- 4 Vapnik-Červonenkis Combinatorics
- 5 Measurability
- 6 Limit Theorems for Vapnik-Červonenkis and Related Classes
- 7 Metric Entropy, with Inclusion and Bracketing
- 8 Approximation of Functions and Sets
- 9 Sums in General Banach Spaces and Invariance Principles
- 10 Universal and Uniform Central Limit Theorems
- 11 The Two-Sample Case, the Bootstrap, and Confidence Sets
- 12 Classes of Sets or Functions Too Large for Central Limit Theorems
- Appendix A Differentiating under an Integral Sign
- Appendix B Multinomial Distributions
- Appendix C Measures on Nonseparable Metric Spaces
- Appendix D An Extension of Lusin's Theorem
- Appendix E Bochner and Pettis Integrals
- Appendix F Nonexistence of Types of Linear Forms on Some Spaces
- Appendix G Separation of Analytic Sets; Borel Injections
- Appendix H Young-Orlicz Spaces
- Appendix I Modifications and Versions of Isonormal Processes
- Subject Index
- Author Index
- Index of Notation
Summary
Let T be any set and (Ω, A, P) a probability space. Recall that a real-valued stochastic process indexed by T is a function (t, ω) ↦ Xt(ω) from T × Ω into ℝ such that for each t ∈ T, Xt (·) is measurable from Ω into ℝ. A modification of the process is another stochastic process Yt defined for the same T and Ω such that for each t, we have P(Xt = Yt) = 1. A version of the process Xt, is a process Zt, t ∈ T, for the same T but defined on a possibly different probability space (Ω1, B, Q) such that Xt and Zt, have the same laws, that is, for each finite subset F of Clearly, any modification of a process is also a version of the process, but a version, even if on the same probability space, may not be a modification. For example, for an isonormal process L on a Hilbert space H, the process M(x) ≔ L(−x) is a version, but not a modification, of L.
One may take a version or modification of a process in order to get better properties such as continuity. It turns out that for the isonormal process on subsets of Hilbert space, what can be done with a version can also be done by a modification, as follows.
TheoremLet L be an isonormal process restricted to a subset C of Hilbert space. For each of the following two properties, if there exists a version M of L with the property, there also is a modification N with the property.
- Type
- Chapter
- Information
- Uniform Central Limit Theorems , pp. 425 - 426Publisher: Cambridge University PressPrint publication year: 1999