At the end of 1960s and the beginning of 1970s, when the Russian version of this book was written, the 'general theory of random processes' did not operate widely with such notions as semimartingale, stochastic integral with respect to semimartingale, the ItO formula for semimartingales, etc. At that time in stochastic calculus (theory of martingales), the main object was the square integrable martingale. In a short time, this theory was applied to such areas as nonlinear filtering, optimal stochastic control, statistics for diffusion­ type processes. In the first edition of these volumes, the stochastic calculus, based on square integrable martingale theory, was presented in detail with the proof of the Doob-Meyer decomposition for submartingales and the description of a structure for stochastic integrals. In the first volume ('General Theory') these results were used for a presentation of further important facts such as the Girsanov theorem and its generalizations, theorems on the innovation pro­ cesses, structure of the densities (Radon-Nikodym derivatives) for absolutely continuous measures being distributions of diffusion and Itô-type processes, and existence theorems for weak and strong solutions of stochastic differential equations. All the results and facts mentioned above have played a key role in the derivation of 'general equations' for nonlinear filtering, prediction, and smoothing of random processes.

Probabilistic and combinatorial techniques are often used for solving advanced problems. This book describes different probabilistic modeling methods and their applications in various areas, such as artificial intelligence, offshore platforms, social networks, and others. It aims to educate how modern probabilistic and combinatorial models may be created to formalize uncertainties; to train how new probabilistic models can be generated for the systems of complex structures; to describe the correct use of the presented models for rational control in systems creation and operation; and to demonstrate analytical possibilities and practical effects for solving different system problems on each life cycle stage.

This text is for an introductory level probability and statistics course with an intermediate algebra prerequisite. The focus of the text follows the American Statistical Association’s Guidelines for Assessment and Instruction in Statistics Education (GAISE). Software examples provided for Microsoft Excel, TI-84 & TI-89 calculators. Students new to statistics are sure to benefit from this ADA accessible and relevant textbook. The examples are current and resonate with everyday life. The casual narrative style, has a conversational tone to provide an inclusive and easy to read format for students.

Computational Genome Analysis: An Introduction presents the foundations of key problems in computational molecular biology and bioinformatics. It focuses on computational and statistical principles applied to genomes, and introduces the mathematics and statistics that are crucial for understanding these applications. The book is appropriate for a one-semester course for advanced undergraduate or beginning graduate students, and it can also introduce computational biology to computer scientists, mathematicians, or biologists who are extending their interests into this exciting field.

This book arose out of two graduate courses that the authors have taught during the past several years the rstone being on measure theoryfollowed by the second one on advanced probability theory. The traditional approach to a ?rst course in measure theory, such as in Royden (1988), is to teach the Lebesgue measure on the real line, then the p di?erentation theorems of Lebesgue, L -spaces on R, and do general m- sure at the end of the course with one main application to the construction of product measures. This approach does have the pedagogic advantage of seeing one concrete case ?rst before going to the general one. But this also has the disadvantage in making many students’ perspective on m- sure theory somewhat narrow. It leads them to think only in terms of the Lebesgue measure on the real line and to believe that measure theory is intimately tied to the topology of the real line. As students of statistics, probability, physics, engineering, economics, and biology know very well, there are mass distributions that are typically nonuniform, and hence it is useful to gain a general perspective. This book attempts to provide that general perspective right from the beginning. The opening chapter gives an informal introduction to measure and integration theory. It shows that the notions of ?-algebra of sets and countable additivity of a set function are dictated by certain very na- ral approximation procedures from practical applications and that they are not just some abstract ideas.

Inference control in statistical databases, also known as statistical disclosure limitation or statistical confidentiality, is about finding tradeoffs to the tension between the increasing societal need for accurate statistical data and the legal and ethical obligation to protect privacy of individuals and enterprises which are the source of data for producing statistics. Techniques used by intruders to make inferences compromising privacy increasingly draw on data mining, record linkage, knowledge discovery, and data analysis and thus statistical inference control becomes an integral part of computer science.

Assuming only calculus and linear algebra, this book introduces the reader in a technically complete way to measure theory and probability, discrete martingales, and weak convergence. It is self- contained and rigorous with a tutorial approach that leads the reader to develop basic skills in analysis and probability. While the original goal was to bring discrete martingale theory to a wide readership, it has been extended so that the book also covers the basic topics of measure theory as well as giving an introduction to the Central Limit Theory and weak convergence. Students of pure mathematics and statistics can expect to acquire a sound introduction to basic measure theory and probability. A reader with a background in finance, business, or engineering should be able to acquire a technical understanding of discrete martingales in the equivalent of one semester. J. C. Taylor is a Professor in the Department of Mathematics and Statistics at McGill University in Montreal. He is the author of numerous articles on potential theory, both probabilistic and analytic, and is particularly interested in the potential theory of symmetric spaces.

This book is an introduction to basic statistical concepts and their applications. The only prerequisite is some knowledge of diffe- rential and integral calculus, which can be picked up in fifteen one-hour lectures. All other mathematical requirements are given in the book itself, in the form of technical notes.