This monograph uses three kinds of probabilistic tools:
The basic results of probability theory, up to the properties of the conditional expectation with respect to a σ-field; they are essentially assumed to be known by the reader but some are briefly recalled in this preliminary chapter.
Some more original expositions of several probabilistic concepts particularly powerful in statistics (such as conditional independence, measurable separability, weak and strong identification among σ-fields, projection of σ-fields). These concepts are presented in separate sections of the chapter in which they are first used. Those probabilistic sections are identified by a slightly different notation; they refer to an abstract probability space ( M , M , P ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9780203758526/6936dfc8-f941-4573-ac88-af4db8cb795d/content/eq221.tif"/> whereas in the statistical sections M is the Cartesian product A × S.
Some more advanced but standard results of probability theory (such as martingale theory, invariance, ergodicity) are used in the last three chapters. These results are not reviewed in this chapter; they are recalled, and where useful reexpressed using our notation, as they become necessary.