Prerequisites from Probability Theory

This introductory chapter contains in particular terms and results from probability theory that are assumed to be known, and which are referred to in the following chapters. These terms include almost sure convergence, convergence in probability, convergence in the p-th mean, and convergence in distribution of real-valued random variables. The reader should be acquainted with basic properties of conditional expectations, the strong law of large numbers, the Borel-Cantelli lemma, and with various inequalities, such as Chebyshev’s, Markov’s, and Jensen’s inequality. Moreover, the central limit theorems of Lindeberg-Lévy and Lindeberg-Feller should be known. The chapter contains a proof of the result that the distribution of a random vector is uniquely determined by its characteristic function. It concludes with basic results from measure and integration theory, such as the monotone convergence theorem, the dominated convergence theorem, the Radon-Nikodým theorem, and Fubini’s theorem.