Over the past thirty years, the development of new areas of Econometrics, such as time series with unit roots characterized by the property of integration, has been based on particularly advanced probabilistic concepts, such as, for example, the functional central limit theorem. The latter is based on the concept of weak convergence of distributions of random functions (stochastic processes), that is, measures of probability defined in the minimal σ-algebra of a specific functional space. In general, the asymptotic properties of econometric estimators and test functions are based on the concept of stochastic convergence, which in turn is based on alternative modes of convergence of sequences of measurable functions.

In addition, the extension of these properties to general cases, where the involved random variables are not independent and identically distributed, has been largely based on concepts that appeared in Probability Theory after 1950, such as martingale and mixing stochastic processes. In addition to Econometrics, Financial Theory has also adopted modern probabilistic concepts in order to develop new methods and models. For example, Brownian motion, random walk, as well as the family of stable limit distributions with infinite variance are concepts widely used in modern Finance. Understanding these concepts requires a satisfactory knowledge of the measure-theoretic approach of Probability Theory. In this approach, probability is treated as a measure defined on a σ-algebra, random variables as measurable functions, and the expected value as a Lebesgue integral. The purpose of this book is to provide the reader with a unified picture of this approach as well as the specific results required for the study of modern Econometrics methods.