Population genetics occupies a central role in a number of important biological and social undertakings. It is fundamental to our understanding of evolutionary processes, of plant and animal breeding programs, and of various diseases of particular importance to mankind.

Our basic question is: Given a collection of DNA sequences, what underlying forces are responsible for the observed patterns of variability? To approach this question we introduce and analyze a number of probability models: the Wright-Fisher model, the coalescent, the infinite alleles model, and the infinite sites model. We study the complications that come from nonconstant population size, recombination, population subdivision, and three forms of natural selection: directional selection, balancing selection, and background selection. These theoretical results set the stage for the investigation of various statistical tests to detect departures from "neutral evolution." The final chapter studies the evolution of whole genomes by chromosomal inversions, reciprocal translocations, and genome duplication.

This text on stochastic processes and their applications is based on a set of lectures given during the past several years at the University of California, Santa Barbara (UCSB). It is an introductory graduate course designed for classroom purposes. Its objective is to provide graduate students of statistics with an overview of some basic methods and techniques in the theory of stochastic processes. The only prerequisites are some rudiments of measure and integration theory and an intermediate course in probability theory. There are more than 50 examples and applications and 243 problems and complements which appear at the end of each chapter. The book consists of 10 chapters. Basic concepts and definitions are pro­ vided in Chapter 1. This chapter also contains a number of motivating ex­ amples and applications illustrating the practical use of the concepts. The last five sections are devoted to topics such as separability, continuity, and measurability of random processes, which are discussed in some detail. The concept of a simple point process on R+ is introduced in Chapter 2. Using the coupling inequality and Le Cam's lemma, it is shown that if its counting function is stochastically continuous and has independent increments, the point process is Poisson. When the counting function is Markovian, the sequence of arrival times is also a Markov process. Some related topics such as independent thinning and marked point processes are also discussed. In the final section, an application of these results to flood modeling is presented.

The word stochastic is jargon for random. A stochastic process is a system which evolves in time while undergoing chance fluctuations. We can describe such a system by defining a family of random variables, {Xt}' where Xt measures, at time t, the aspect of the system which is of interest. For example, X t might be the number of customers in a queue at time t. As time passes, customers will arrive and leave, and so the value of Xt will change. At any time t, Xt takes one of the values 0, 1,2, … ; and t can be any value in a subset of ( - 00, 00), the infinite past to the infinite future. If we observe the queue continuously, and customers arrive one at a time to be served by a single server, then, when a customer arrives, the value of Xt' the queue size, increases by one, and when a customer departs after being served, X t decreases by one (Figure 1.1).

"Probability and Partial Differential Equations in Modern Applied Mathematics" is devoted to the role of probabilistic methods in modern applied mathematics from the perspectives of both a tool for analysis and as a tool in modeling. There is a recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes.

This book presents applied probability and stochastic processes in an elementary but mathematically precise manner, with numerous examples and exercises to illustrate the range of engineering and science applications of the concepts. The book is designed to give the reader an intuitive understanding of probabilistic reasoning, in addition to an understanding of mathematical concepts and principles. The initial chapters present a summary of probability and statistics and then Poisson processes, Markov chains, Markov processes and queuing processes are introduced. Advanced topics include simulation, inventory theory, replacement theory, Markov decision theory, and the use of matrix geometric procedures in the analysis of queues.

In this monograph the authors develop a theory for the robust control of discrete-time stochastic systems, subjected to both independent random perturbations and to Markov chains. Such systems are widely used to provide mathematical models for real processes in fields such as aerospace engineering, communications, manufacturing, finance and economy. The theory is a continuation of the authors’ work presented in their previous book entitled "Mathematical Methods in Robust Control of Linear Stochastic Systems" published by Springer in 2006.

In the Preface to the first edition, originally published in 1980, we mentioned that this book was based on the author's lectures in the Department of Mechanics and Mathematics of the Lomonosov University in Moscow, which were issued, in part, in mimeographed form under the title "Probabil­ ity, Statistics, and Stochastic Processors, I, II" and published by that Univer­ sity. Our original intention in writing the first edition of this book was to divide the contents into three parts: probability, mathematical statistics, and theory of stochastic processes, which corresponds to an outline of a three­ semester course of lectures for university students of mathematics. However, in the course of preparing the book, it turned out to be impossible to realize this intention completely, since a full exposition would have required too much space.

Most probability problems involve random variables indexed by space and/or time. These problems almost always have a version in which space and/or time are taken to be discrete. This volume deals with areas in which the discrete version is more natural than the continuous one, perhaps even the only one than can be formulated without complicated constructions and machinery.

Statistics and Analysis of Scientific Data covers the foundations of probability theory and statistics, and a number of numerical and analytical methods that are essential for the present-day analyst of scientific data. Topics covered include probability theory, distribution functions of statistics, fits to two-dimensional datasheets and parameter estimation, Monte Carlo methods and Markov chains. Equal attention is paid to the theory and its practical application, and results from classic experiments in various fields are used to illustrate the importance of statistics in the analysis of scientific data.

An outgrowth of the 12th International Conference on Applied Stochastic Models and Data Analysis, this book is a collection of invited chapters presenting recent developments in the field of data analysis, with applications to reliability and inference, data mining, bioinformatics, lifetime data, and neural networks. Emphasized throughout the volume are new methods with the potential for solving real-world problems in various areas.

This book is a collection of exercises covering all the main topics in the modern theory of stochastic processes and its applications, including finance, actuarial mathematics, queuing theory, and risk theory.

Changes in the second edition. The second edition differs from the first in that there is a full development of problems where the variance of the diffusion term and the jump distribution can be controlled. Also, a great deal of new material concerning deterministic problems has been added, including very efficient algorithms for a class of problems of wide current interest.

With the first edition out of print, we decided to arrange for republi­ cation of Denumerrible Markov Ohains with additional bibliographic material. The new edition contains a section Additional Notes that indicates some of the developments in Markov chain theory over the last ten years.

Stochastic processes are necessary ingredients for building models of a wide variety of phenomena exhibiting time varying randomness. In a lively and imaginative presentation, studded with examples, exercises, and applications, and supported by inclusion of computational procedures, the author has created a textbook that provides easy access to this fundamental topic for many students of applied sciences at many levels. With its carefully modularized discussion and crystal clear differentiation between rigorous proof and plausibility argument, it is accessible to beginners but flexible enough to serve as well those who come to the course with strong backgrounds.