Ebook: Probability

This book contains a systematic treatment of probability from the ground up, starting with intuitive ideas and gradually developing more sophisticated subjects, such as random walks, martingales, Markov chains, ergodic theory, weak convergence of probability measures, stationary stochastic processes, and the Kalman-Bucy filter. Many examples are discussed in detail, and there are a large number of exercises. The book is accessible to advanced undergraduates and can be used as a text for self-study.;Ch. I. Elementary Probability Theory -- Ch. II. Mathematical Foundations of Probability Theory -- Ch. III. Convergence of Probability Measures. Central Limit Theorem -- Ch. IV. Sequences and Sums of Independent Random Variables -- Ch. V. Stationary (Strict Sense) Random Sequences and Ergodic Theory -- Ch. VI. Stationary (Wide Sense) Random Sequences. L[superscript 2] Theory -- Ch. VII. Sequences of Random Variables that Form Martingales -- Ch. VIII. Sequences of Random Variables that Form Markov Chains.;This new edition contains substantial revisions and updated references. The reader will find a deeper study of topics such as the distance between probability measures, metrization of weak convergence, and contiguity of probability measures. Proofs for a number of some important results which were merely stated in the first edition have been added. The author has included new material on the probability of large deviations, on the central limit theorem for sums of dependent random variables, and on a discrete version of Ito's formula.