Probabilistic Inequalities.
Title:
Probabilistic Inequalities.
Author:
Anastassiou, George A.
ISBN:
9789814280792
Personal Author:
Physical Description:
1 online resource (429 pages)
Series:
Series on Concrete and Applicable Mathematics ; v.7
Series on Concrete and Applicable Mathematics
Contents:
Contents -- Preface -- 1. Introduction -- 2. Basic Stochastic Ostrowski Inequalities -- 2.1 Introduction -- 2.2 One Dimensional Results -- 2.3 Multidimensional Results -- 2.4 Applications -- 2.5 Addendum -- 3. Multidimensional Montgomery Identities and Ostrowski type Inequalities -- 3.1 Introduction -- 3.2 Results -- 3.3 Application to Probability Theory -- 4. General Probabilistic Inequalities -- 4.1 Introduction -- 4.2 Make Applications -- 4.3 Remarks on an Inequality -- 4.4 Lq -- q > 1, Related Theory -- 5. About Grothendieck Inequalities -- 5.1 Introduction -- 5.2 Main Results -- 6. Basic Optimal Estimation of Csiszar's f-Divergence -- 6.1 Background -- 6.2 Main Results -- 7. Approximations via Representations of Csiszar's f-Divergence -- 7.1 Background -- 7.2 All Results -- 8. Sharp High Degree Estimation of Csiszar's f-Divergence -- 8.1 Background -- 8.2 Results Based on Taylor's Formula -- 8.3 Results Based on Generalized Taylor{Widder Formula -- 8.4 Results Based on an Alternative Expansion Formula -- 9. Csiszar's f-Divergence as a Measure of Dependence -- 9.1 Background -- 9.2 Results -- 10. Optimal Estimation of Discrete Csiszar f-Divergence -- 10.1 Background -- 10.2 Results -- 11. About a General Discrete Measure of Dependence -- 11.1 Background -- 11.2 Results -- 12. H older-Like Csiszar's f-Divergence Inequalities -- 12.1 Background -- 12.2 Results -- 13. Csiszar's Discrimination and Ostrowski Inequalities via Eulertype and Fink Identities -- 13.1 Background -- 13.2 Main Results -- 14. Taylor-Widder Representations and Gr uss, Means, Ostrowski and Csiszar's Inequalities -- 14.1 Introduction -- 14.2 Background -- 14.3 Main Results -- 15. Representations of Functions and Csiszar's f-Divergence -- 15.1 Introduction -- 15.2 Main Results -- 16. About General Moment Theory -- 16.1 The Standard Moment Problem.
16.2 The Convex Moment Problem -- 16.3 Infinite Many Conditions Moment Problem -- 16.4 Applications - Discussion -- 17. Extreme Bounds on the Average of a Rounded o Observation under a Moment Condition -- 17.1 Preliminaries -- 17.2 Results -- 18. Moment Theory of Random Rounding Rules Subject to One Moment Condition -- 18.1 Preliminaries -- 18.2 Bounds for Random Rounding Rules -- 18.3 Comparison of Bounds for Random and Deterministic Rounding Rules -- 19. Moment Theory on Random Rounding Rules Using Two Moment Conditions -- 19.1 Preliminaries -- 19.2 Results -- 19.3 Proofs -- 20. Prokhorov Radius Around Zero Using Three Moment Constraints -- 20.1 Introduction and Main Result -- 20.2 Auxiliary Moment Problem -- 20.3 Proof of Theorem 20.1. -- 20.4 Concluding Remarks -- 21. Precise Rates of Prokhorov Convergence Using Three Moment Conditions -- 21.1 Main Result -- 21.2 Outline of Proof -- 22. On Prokhorov Convergence of Probability Measures to the Unit under Three Moments -- 22.1 Main Result -- 22.2 An Auxiliary Moment Problem -- 22.3 Further Auxiliary Results -- 22.4 Minimizing Solutions in Boxes -- 22.5 Conclusions -- 23. Geometric Moment Methods Applied to Optimal Portfolio Management -- 23.1 Introduction -- 23.2 Preliminaries -- 23.3 Main Results -- 23.3.1. The Financial Portfolio Problem -- 23.3.2. The Portfolio Problem Involving Three Securities (similarly one can treat the problem of more than three securities). -- 23.3.3. Applications of -Moment Problem to Variance -- 23.3.4. The Multivariate Case -- 24. Discrepancies Between General Integral Means -- 24.1 Introduction -- 24.2 Results -- 25. Gruss Type Inequalities Using the Stieltjes Integral -- 25.1 Motivation -- 25.2 Main Results -- 25.3 Applications -- 26. Chebyshev-Gr uss Type and Di erence of Integral Means Inequalities Using the Stieltjes Integral -- 26.1 Background -- 26.2 Main Results.
26.3 Applications -- 27. An Expansion Formula -- 27.1 Results -- 27.2 Applications -- 28. Integration by Parts on the Multidimensional Domain -- 28.1 Results -- Bibliography -- List of Symbols -- Index.
Abstract:
In this monograph, the author presents univariate and multivariate probabilistic inequalities with coverage on basic probabilistic entities like expectation, variance, moment generating function and covariance. These are built on the recent classical form of real analysis inequalities which are also discussed in full details. This treatise is the culmination and crystallization of the author's last two decades of research work in related discipline. Each of the chapters is self-contained and a few advanced courses can be taught out of this book. Extensive background and motivations for specific topics are given in each chapter. A very extensive list of references is also provided at the end. The topics covered in this unique book are wide-ranging and diverse. The opening chapters examine the probabilistic Ostrowski type inequalities, and various related ones, as well as the largely discusses about the Grothendieck type probabilistic inequalities. The book is also about inequalities in information theory and the Csiszar's f -Divergence between probability measures. A great section of the book is also devoted to the applications in various directions of Geometry Moment Theory. Also, the development of the Grüss type and Chebyshev-Grüss type inequalities for Stieltjes integrals and the applications in probability are explored in detail. The final chapters discuss the important real analysis methods with potential applications to stochastics. The book will be of interest to researchers and graduate students, and it is also seen as an invaluable reference book to be acquired by all science libraries as well as seminars that conduct discussions on related topics. Sample Chapter(s). Chapter 1: Introduction (59 KB). Contents: Basic Stochastic Ostrowski Inequalities; General Probabilistic Inequalities; Optimal Estimation of Discrete Csiszar f -Divergence;
Hölder-Like Csiszar's f -Divergence Inequalities; Prokhorov Radius Around Zero Using Three Moment Constraints; Precise Rates of Prokhorov Convergence Using Three Moment Conditions; Discrepancies Between General Integral Means; An Expansion Formula; and other chapters. Readership: Graduate students and researchers in pure mathematics, applied mathematics, probability and statistics.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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