Kappos / Birnbaum / Lukacs Probability Algebras and Stochastic Spaces

1. Auflage 2014
ISBN: 978-1-4832-1850-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 280 Seiten, Web PDF

Probability Algebras and Stochastic Spaces explores the fundamental notions of probability theory in the so-called 'point-free way. The space of all elementary random variables defined over a probability algebra in a 'point-free way is a base for the stochastic space of all random variables, which can be obtained from it by lattice-theoretic extension processes. This book is composed of eight chapters and begins with discussions of the definition, properties, scope, and extension of probability algebras. The succeeding chapters deal with the Cartesian product of probability algebras and the principles of stochastic spaces. These topics are followed by surveys of the expectation, moments, and spaces of random variables. The final chapters define generalized random variables and the Boolean homomorphisms of these variables. This book will be of great value to mathematicians and advance mathematics students.

Autoren/Hrsg.

Kappos, Demetrios A.

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Probability Algebras and Stochastic Spaces;4
3;Copyright Page;5
4;Table of Contents;10
5;Preface;6
6;Chapter I. Probability algebras;14
6.1;1. Definition and properties;14
6.2;2. Probability subalgebras;16
6.3;3. Isometric probability algebras;16
6.4;4. Examples;16
6.5;5. Separability relative to a probability;20
6.6;6. Countably additive [s-additive]
probabilities;20
6.7;7. Probability
s-algebras;23
6.8;8. Quasi-probability algebras;25
6.9;9. Probability spaces;26
7;Chapter II. Extension of probability algebras;29
7.1;1.
The probability algebra as a metric space;29
7.2;2. Construction of a
s-extension of a probability algebra;31
7.3;3. The linear Lebesgue probability
s-algebra;42
7.4;4. Classification of the p-separable probability
s-algebras;46
8;Chapter III. Cartesian product of probability algebras;48
8.1;1. Cartesian product of Boolean alegbras;48
8.2;2. Product probability algebras;53
8.3;3. Classification of probability
s-algebras;59
8.4;4. Representation of probability
s-algebras by probability spaces;64
8.5;5. Independence in probability;66
9;Chapter IV. Stochastic spaces;68
9.1;1. Experiments (trials) in probability algebras;68
9.2;2. Elementary random variables (elementary stochastic space);71
9.3;3. Convergence in stochastic spaces;80
9.4;4. o-convergence in .
with respect to a vector sublattice of .;89
9.5;5. Extension of the elementary stochastic space;102
9.6;6. Stochastic space of all bounded random variables [rv'
s];109
9.7;7. Convergence in probability and almost uniform convergence;110
9.8;8. Generators of the stochastic space;116
9.9;9. Other convergences in the stochastic space;119
9.10;10. Closure operator in the stochastic space;122
9.11;11. Series convergence;123
9.12;12. Composition of random variables;133
10;Chapter V. Expectation of random variables;135
10.1;1. Expectation of elementary random variables;135
10.2;2. The space L1 of all RV'S with
expectation;143
10.3;3. Signed measures;151
10.4;4. The Radon–Nikodym
Theorem;155
10.5;5. Remarks;157
11;Chapter VI. Moments. Spaces
Lq;161
11.1;1. Powers of rv's;161
11.2;2. Moments of random variables;162
11.3;3. The spaces
Lq;164
11.4;4. Convergence in mean and equi-integrability;167
12;Chapter VII. Generalized random variables (Random variables having values in any space);174
12.1;1. Preliminaries;174
12.2;2. Generalized elementary random variables;175
12.3;3. Completion with respect to o-convergence;176
12.4;4. Completion with respect to a norm;192
12.5;5. Expectation of rv's having values in a Banach space;200
12.6;6. The spaces
Lq of rv's having values in a Banach space. Moments;225
13;Chapter VIII. Complements;237
13.1;1. The Radon–Nikodym theorem for
the Bochner integral;237
13.2;2. Conditional probability;239
13.3;3. Conditional expectation;241
13.4;4. Distributions of random variables;243
13.5;5. Boolean homomorphisms of
rv's;245
14;Appendix I: Lattices;246
14.1;1. Partially ordered sets;246
14.2;2. Lattices;247
14.3;3. Boolean algebras;250
14.4;4. Homomorphisms and ideals of a Boolean algebra;255
14.5;5. Order convergence;260
14.6;6. Closures;261
15;Appendix II: Lattice groups, vector lattices;263
15.1;1. Lattice groups;263
15.2;2. Vector lattices or linear lattice spaces;266
16;Bibliographical Notes;268
17;Bibliography;270
18;List of Symbols;274
19;Index;276