Cuppens / Birnbaum / Lukacs Decomposition of Multivariate Probabilities

1. Auflage 2014
ISBN: 978-1-4832-1764-2
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 262 Seiten, Web PDF

Decomposition of Multivariate Probability is a nine-chapter text that focuses on the problem of multivariate characteristic functions. After a brief introduction to some useful results on measures and integrals, this book goes on dealing with the classical theory and the Fourier-Stieltjes transforms of signed measures. The succeeding chapters explore the multivariate extension of the well-known Paley-Wiener theorem on functions that are entire of exponential type and square-integrable; the theory of infinitely divisible probabilities and the classical results of Hin?in; and the decompositions of analytic characteristic functions. Other chapters are devoted to the important problem of the description of a specific class on n-variate probabilities without indecomposable factors. The final chapter studies the problem of ?-decomposition of multivariate characteristic functions. This book will prove useful to mathematicians and advance undergraduate and graduate students.

Autoren/Hrsg.

Cuppens, Roger

Birnbaum, Z. W.

Lukacs, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Decomposition of Multivariate Probabilities;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Notation;12
7;List of Symbols;16
8;Chapter 1. Measures and Integrals;18
8.1;1.1 Measures;18
8.2;1.2 Integrals;20
8.3;1.3 Product Measures;22
8.4;1.4 Signed Measures;23
8.5;1.5 Singular and Absolutely Continuous Measures;25
8.6;1.6 Continuity Sets;26
9;Chapter 2. Fourier-Stieltjes Transforms of Signed Measures;28
9.1;2.1 Fourier-Stieltjes Transforms;28
9.2;2.2 Uniqueness Theorem;30
9.3;2.3 Inversion Formulas;33
9.4;2.4 Projection Theorem;36
9.5;2.5 Convolution Theorem;38
9.6;2.6 Continuity Theorems;42
9.7;2.7 Bochner's Theorem;57
9.8;2.8 A Characterization of the Fourier-Stieltjes Transform;59
9.9;2.9 Exponential of Signed Measures;64
9.10;Notes;66
10;Chapter 3. Analytic Characteristic Functions;67
10.1;3.1 Examples of Characteristic Functions;67
10.2;3.2 Derivatives of Characteristic Functions;69
10.3;3.3 Analytic Characteristic Functions;71
10.4;3.4 Some Characterization Theorems;78
10.5;3.5 An Extension of the Notion of Analytic Characteristic Functions;83
10.6;3.6 Convex Support of Signed Measures;85
10.7;Notes;90
11;Chapter 4. Decomposition Theorems;92
11.1;4.1 Indecomposable Probabilities;92
11.2;4.2 Infinitely Divisible Probabilities;95
11.3;4.3 Canonical Representations;99
11.4;4.4 A Limit Theorem;114
11.5;4.5 Hincin's Theorem;119
11.6;4.6 Probabilities with No Indecomposable Factor;123
11.7;Notes;126
12;Chapter 5. Decomposition Theorems for Analytic Characteristic Functions;128
12.1;5.1 Decompositions of Derivable Characteristic Functions;128
12.2;5.2 Decompositions of Probabilities Belonging to Ar;130
12.3;5.3 Decompositions of Analytic Characteristic Functions;134
12.4;Notes;138
13;Chapter 6. Infinitely Divisible Probabilities with Normal Factor;139
13.1;6.1 Case n = 1;139
13.2;6.2 A Necessary Condition;140
13.3;6.3 Induction Method;141
13.4;6.4 Some Sufficient Conditions for Membership to Ino;156
13.5;Notes;163
14;Chapter 7. Infinitely Divisible Probabilities without Normal Factor;164
14.1;7.1 Probabilities with a Poisson Measure Concentrated on a Strip;164
14.2;7.2 Probabilities Having an Absolutely Continuous Poisson Measure;174
14.3;7.3 Isomorphism Method;177
14.4;7.4 Independent Sets;181
14.5;7.5 Independent Sets and Projections;185
14.6;Notes;188
15;Chapter 8. Infinitely Divisible Probabilities with Countable Poisson Spectrum;190
15.1;8.1 The General Case;190
15.2;8.2 Lattice Probabilities;193
15.3;8.3 Extensions to Independent Sets;202
15.4;8.4 Finite Products of Poisson Probabilities;204
15.5;Notes;215
16;Chapter 9. a-Decomposition;216
16.1;9.1 Statement of the Problem;216
16.2;9.2 a-Decompositions of Probabilities with Analytic Characteristic Functions;218
16.3;9.3 Probabilities without Indecomposable a-Factors;220
16.4;Notes;221
17;Appendix A: Some Results of Function Theory;222
17.1;A.1 Stone-Weierstrass Theorem;222
17.2;A.2 Almost Periodic Functions;223
17.3;A.3 Independent Sets;225
17.4;A.4 Analytic Functions;229
17.5;A.5 Topologically Independent Functions;231
18;Appendix B: Exponentials of Polynomials and Functions;234
18.1;B.1 Case of a Polynomial of One Variable;234
18.2;B.2 Case of a Function of One Variable;246
18.3;B.3 Case of Functions of Several Variables;251
19;References;253
20;Index;260