Decomposition of Multivariate Probabilities

Preface

Notation

List of Symbols

Chapter 1 Measures and Integrals


1.1 Measures


1.2 Integrals


1.3 Product Measures


1.4 Signed Measures


1.5 Singular and Absolutely Continuous Measures


1.6 Continuity Sets

Chapter 2 Fourier-Stieltjes Transforms of Signed Measures


2.1 Fourier-Stieltjes Transforms


2.2 Uniqueness Theorem


2.3 Inversion Formulas


2.4 Projection Theorem


2.5 Convolution Theorem


2.6 Continuity Theorems


2.7 Bochner's Theorem


2.8 A Characterization of the Fourier-Stieltjes Transform


2.9 Exponential of Signed Measures

Notes

Chapter 3 Analytic Characteristic Functions


3.1 Examples of Characteristic Functions


3.2 Derivatives of Characteristic Functions


3.3 Analytic Characteristic Functions


3.4 Some Characterization Theorems


3.5 An Extension of the Notion of Analytic Characteristic Functions


3.6 Convex Support of Signed Measures

Notes

Chapter 4 Decomposition Theorems


4.1 Indecomposable Probabilities


4.2 Infinitely Divisible Probabilities


4.3 Canonical Representations


4.4 A Limit Theorem


4.5 Hinčin's Theorem


4.6 Probabilities with No Indecomposable Factor

Notes

Chapter 5 Decomposition Theorems for Analytic Characteristic Functions


5.1 Decompositions of Derivable Characteristic Functions


5.2 Decompositions of Probabilities Belonging to Ar


5.3 Decompositions of Analytic Characteristic Functions

Notes

Chapter 6 Infinitely Divisible Probabilities with Normal Factor


6.1 Case n = 1


6.2 A Necessary Condition


6.3 Induction Method


6.4 Some Sufficient Conditions for Membership to In0

Notes

Chapter 7 Infinitely Divisible Probabilities without Normal Factor


7.1 Probabilities with a Poisson Measure Concentrated on a Strip


7.2 Probabilities Having an Absolutely Continuous Poisson Measure


7.3 Isomorphism Method


7.4 Independent Sets


7.5 Independent Sets and Projections

Notes

Chapter 8 Infinitely Divisible Probabilities with Countable Poisson Spectrum


8.1 The General Case


8.2 Lattice Probabilities


8.3 Extensions to Independent Sets


8.4 Finite Products of Poisson Probabilities

Notes

Chapter 9 α-Decomposition


9.1 Statement of the Problem


9.2 α-Decompositions of Probabilities with Analytic Characteristic Functions


9.3 Probabilities without Indecomposable α-Factors

Notes

Appendix A Some Results of Function Theory

A.1 Stone-Weierstrass Theorem

A.2 Almost Periodic Functions

A.3 Independent Sets

A.4 Analytic Functions

A.5 Topologically Independent Functions

Appendix B Exponentials of Polynomials and Functions

B.1 Case of a Polynomial of One Variable

B.2 Case of a Function of One Variable

B.3 Case of Functions of Several Variables

References

Index