Inequalities and Extremal Problems in Probability and Statistics: Selected Topics presents various kinds of useful inequalities that are applicable in many areas of mathematics, the sciences, and engineering. The book enables the reader to grasp the importance of inequalities and how they relate to probability and statistics.
This will be an extremely useful book for researchers and graduate students in probability, statistics, and econometrics, as well as specialists working across sciences, engineering, financial mathematics, insurance, and mathematical modeling of large risks.
1. 1.1 Introduction - p. 1 2. 1.2 An Inequality for the Martingale Difference Sequence - p. 7 3. 1.3 An Inequality for the Martingale Square Function - p. 12 4. 1.4 A Prophet Inequality for L2-Bounded Martingales - p. 15 5. 1.5 Moment Inequality for Martingale Square Function - p. 19 6. 1.6 An Inequality for Martingale Transforms - p. 23 7. Chapter 2: On a Class of Optimal Stopping Problems of Nonintegral Type - p. 27 8. Chapter 3: On the Absolute Constants in Nagaev–Bikelis-Type Inequalities - p. 45 9. Abstract - p. 47 10. Acknowledgments - p. 47 11. 3.1 Introduction and Formulation of the Main Results - p. 47 12. 3.2 Auxiliary Statements - p. 60 13. 3.3 Cases (i) and (iii)—“small” and “large” x - p. 63 14. 3.4 Case (ii)—“moderate” x - p. 66 15. Chapter 4: On the Nonuniform Berry–Esseen Bound - p. 102 16. Abstract - p. 103 17. 4.1 Uniform and Nonuniform Berry–Esseen (BE) Bounds - p. 103 18. 4.2 The Bohman–Prawitz–Vaaler Smoothing Inequalities - p. 104 19. 4.3 Nonuniform BE Bounds: Nagaev’s Result and Method - p. 106 20. 4.4 A New Way to Obtain Nonuniform BE Bounds - p. 114 21. 4.5 Constructions of the Smoothing Filter M - p. 122 22. 4.6 Another Construction of Smoothing Inequalities for Nonuniform BE Bounds - p. 129 23. Chapter 5: On the Berry–Esseen Bound for the Student Statistic - p. 138 24. Chapter 6: Sharp Probability Inequalities for Random Polynomials, Generalized Sample Cross-Moments, and Studentized Processes - p. 157 25. Abstract - p. 159 26. Acknowledgments - p. 159 27. 6.1 Introduction - p. 159 28. 6.2 Sharp Probability Inequalities for Sums of Independent R.V.’s and Their Self-Normalized Analogs - p. 161 29. 6.3 Main Inequalities - p. 164 30. 6.4 Applications in Hypothesis Testing - p. 173 31. 6.5 Applications in Testing Procedures for Dependent R.V.’s - p. 176 32. 6.6 Proofs - p. 179
Inequalities and Extremal Problems in Probability and Statistics: Selected Topics presents various kinds of useful inequalities that are applicable in many areas of mathematics, the sciences, and engineering. The book enables the reader to grasp the importance of inequalities and how they relate to probability and statistics.
This will be an extremely useful book for researchers and graduate students in probability, statistics, and econometrics, as well as specialists working across sciences, engineering, financial mathematics, insurance, and mathematical modeling of large risks.
1. 1.1 Introduction - p. 1 2. 1.2 An Inequality for the Martingale Difference Sequence - p. 7 3. 1.3 An Inequality for the Martingale Square Function - p. 12 4. 1.4 A Prophet Inequality for L2-Bounded Martingales - p. 15 5. 1.5 Moment Inequality for Martingale Square Function - p. 19 6. 1.6 An Inequality for Martingale Transforms - p. 23 7. Chapter 2: On a Class of Optimal Stopping Problems of Nonintegral Type - p. 27 8. Chapter 3: On the Absolute Constants in Nagaev–Bikelis-Type Inequalities - p. 45 9. Abstract - p. 47 10. Acknowledgments - p. 47 11. 3.1 Introduction and Formulation of the Main Results - p. 47 12. 3.2 Auxiliary Statements - p. 60 13. 3.3 Cases (i) and (iii)—“small” and “large” x - p. 63 14. 3.4 Case (ii)—“moderate” x - p. 66 15. Chapter 4: On the Nonuniform Berry–Esseen Bound - p. 102 16. Abstract - p. 103 17. 4.1 Uniform and Nonuniform Berry–Esseen (BE) Bounds - p. 103 18. 4.2 The Bohman–Prawitz–Vaaler Smoothing Inequalities - p. 104 19. 4.3 Nonuniform BE Bounds: Nagaev’s Result and Method - p. 106 20. 4.4 A New Way to Obtain Nonuniform BE Bounds - p. 114 21. 4.5 Constructions of the Smoothing Filter M - p. 122 22. 4.6 Another Construction of Smoothing Inequalities for Nonuniform BE Bounds - p. 129 23. Chapter 5: On the Berry–Esseen Bound for the Student Statistic - p. 138 24. Chapter 6: Sharp Probability Inequalities for Random Polynomials, Generalized Sample Cross-Moments, and Studentized Processes - p. 157 25. Abstract - p. 159 26. Acknowledgments - p. 159 27. 6.1 Introduction - p. 159 28. 6.2 Sharp Probability Inequalities for Sums of Independent R.V.’s and Their Self-Normalized Analogs - p. 161 29. 6.3 Main Inequalities - p. 164 30. 6.4 Applications in Hypothesis Testing - p. 173 31. 6.5 Applications in Testing Procedures for Dependent R.V.’s - p. 176 32. 6.6 Proofs - p. 179
