Review by Choice Review
Roussas (Univ. of California, Davis) provides basic tools in measure theory and probability, in the classical spirit, relying heavily on characteristic functions as tools without using martingale or empirical process methods. Readers need good backgrounds in real analysis. Chapters (15) discuss field, sigma-field, monotone class, measure, outer measure, various types of convergence; integral of a random variable (r.v.) x; probability distribution of x; standard convergence theorems such as the Lebesgue Monotone Convergence Theorem, the Dominated Convergence Theorem, the Fubini Theorem; various probability inequalities (such as Holder inequality, Jensen inequality); convergence in the r th mean; three important theorems (Hahn-Jordan Decomposition Theorem, Lebesgue Decomposition Theorem, Radon-Nikodyn Theorem); distribution functions; some Helly-Bray type theorems; conditional expectations and conditional probabilities in terms of sigma-fields; weak and complete convergence of a sequence of distribution functions; independence of random variables and consequences; characteristic functions as powerful tools; obtaining conditions under which partial sums of independent random variables converge in distribution to some limiting law (e.g., normal distribution); sequences of random variables independent but not identically distributed; Strong Law of Large Numbers; and ergodic theory. A useful appendix overviews each chapter. A well-written book. Chapter exercise sets. ^BSumming Up: Highly recommended. Graduate students; faculty. D. V. Chopra Wichita State University
Copyright American Library Association, used with permission.
Review by Choice Review
