Main Directions in the Theory of Probability Metrics

Abstract

Increasingly, the demands of various real-world applications in the sciences, engineering, and business have resulted in the creation of new, more complicated probability models. In the construction and evaluation of these models, model builders have drawn on well-developed limit theorems in probability theory and the theory of random processes. The study of limit theorems in general spaces and a number of other questions in probability theory make it necessary to introduce functionals – defined on either classes of probability distributions or classes of random elements – and to evaluate their nearness in one or another probabilistic sense. Thus various metrics have appeared including the well-known Kolmogorov (uniform) metric, L _p metrics, the Prokhorov metric, and the metric of convergence in probability (Ky Fan metric). We discuss these measures and others in the chapters that follow.