Ebook: The Methods of Distances in the Theory of Probability and Statistics

This book covers the method of metric distances and its application in probability theory and other fields. The method is fundamental in the study of limit theorems and generally in assessing the quality of approximations to a given probabilistic model. The method of metric distances is developed to study stability problems and reduces to the selection of an ideal or the most appropriate metric for the problem under consideration and a comparison of probability metrics.

After describing the basic structure of probability metrics and providing an analysis of the topologies in the space of probability measures generated by different types of probability metrics, the authors study stability problems by providing a characterization of the ideal metrics for a given problem and investigating the main relationships between different types of probability metrics. The presentation is provided in a general form, although specific cases are considered as they arise in the process of finding supplementary bounds or in applications to important special cases.

Svetlozar T. Rachev is the Frey Family Foundation Chair of Quantitative Finance, Department of Applied Mathematics and Statistics, SUNY-Stony Brook and Chief Scientist of Finanlytica, USA. Lev B. Klebanov is a Professor in the Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic. Stoyan V. Stoyanov is a Professor at EDHEC Business School and Head of Research, EDHEC-Risk Institute—Asia (Singapore). Frank J. Fabozzi is a Professor at EDHEC Business School. (USA)

This book covers the method of metric distances and its application in probability theory and other fields. The method is fundamental in the study of limit theorems and generally in assessing the quality of approximations to a given probabilistic model. The method of metric distances is developed to study stability problems and reduces to the selection of an ideal or the most appropriate metric for the problem under consideration and a comparison of probability metrics.

After describing the basic structure of probability metrics and providing an analysis of the topologies in the space of probability measures generated by different types of probability metrics, the authors study stability problems by providing a characterization of the ideal metrics for a given problem and investigating the main relationships between different types of probability metrics. The presentation is provided in a general form, although specific cases are considered as they arise in the process of finding supplementary bounds or in applications to important special cases.

Svetlozar T. Rachev is the Frey Family Foundation Chair of Quantitative Finance, Department of Applied Mathematics and Statistics, SUNY-Stony Brook and Chief Scientist of Finanlytica, USA. Lev B. Klebanov is a Professor in the Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic. Stoyan V. Stoyanov is a Professor at EDHEC Business School and Head of Research, EDHEC-Risk Institute—Asia (Singapore). Frank J. Fabozzi is a Professor at EDHEC Business School. (USA)

This book covers the method of metric distances and its application in probability theory and other fields. The method is fundamental in the study of limit theorems and generally in assessing the quality of approximations to a given probabilistic model. The method of metric distances is developed to study stability problems and reduces to the selection of an ideal or the most appropriate metric for the problem under consideration and a comparison of probability metrics.

After describing the basic structure of probability metrics and providing an analysis of the topologies in the space of probability measures generated by different types of probability metrics, the authors study stability problems by providing a characterization of the ideal metrics for a given problem and investigating the main relationships between different types of probability metrics. The presentation is provided in a general form, although specific cases are considered as they arise in the process of finding supplementary bounds or in applications to important special cases.

Svetlozar T. Rachev is the Frey Family Foundation Chair of Quantitative Finance, Department of Applied Mathematics and Statistics, SUNY-Stony Brook and Chief Scientist of Finanlytica, USA. Lev B. Klebanov is a Professor in the Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic. Stoyan V. Stoyanov is a Professor at EDHEC Business School and Head of Research, EDHEC-Risk Institute—Asia (Singapore). Frank J. Fabozzi is a Professor at EDHEC Business School. (USA)

Content:
Front Matter....Pages i-xvi
Front Matter....Pages 9-9
Probability Distances and Probability Metrics: Definitions....Pages 11-31
Primary, Simple, and Compound Probability Distances and Minimal and Maximal Distances and Norms....Pages 33-66
A Structural Classification of Probability Distances....Pages 67-105
Front Matter....Pages 107-107
Monge–Kantorovich Mass Transference Problem, Minimal Distances and Minimal Norms....Pages 109-143
Quantitative Relationships Between Minimal Distances and Minimal Norms....Pages 145-167
Relations Between Minimal and Maximal Distances....Pages 169-197
Moment Problems Related to the Theory of Probability Metrics: Relations Between Compound and Primary Distances....Pages 199-217
Front Matter....Pages 219-233
Moment Distances....Pages 235-235
Uniformity in Weak and Vague Convergence....Pages 237-270
Glivenko–Cantelli Theorem and Bernstein–Kantorovich Invariance Principle....Pages 271-282
Stability of Queueing Systems....Pages 283-296
Optimal Quality Usage....Pages 297-315
Front Matter....Pages 317-331
Ideal Metrics with Respect to Summation Scheme for i.i.d. Random Variables....Pages 333-333
Ideal Metrics and Rate of Convergence in the CLT for Random Motions....Pages 335-362
Applications of Ideal Metrics for Sums of i.i.d. Random Variables to the Problems of Stability and Approximation in Risk Theory....Pages 363-378
How Close Are the Individual and Collective Models in Risk Theory?....Pages 379-393
Ideal Metric with Respect to Maxima Scheme of i.i.d. Random Elements....Pages 395-420
Ideal Metrics and Stability of Characterizations of Probability Distributions....Pages 421-477
Main Directions in the Theory of Probability Metrics....Pages 479-516
Front Matter....Pages 1-7
Positive and Negative Definite Kernels and Their Properties....Pages 517-517
Negative Definite Kernels and Metrics: Recovering Measures from Potentials....Pages 519-537
Statistical Estimates Obtained by the Minimal Distances Method....Pages 539-569
Distances Defined by Zonoids....Pages 571-579
Back Matter....Pages 581-588
....Pages 589-597