Robert B. Ash

This introduction to more advanced courses in probability and real analysis emphasizes the probabilistic way of thinking, rather than measure-theoretic concepts. Geared toward advanced undergraduates and graduate students, its sole prerequisite is calculus. Taking statistics as its major field of application, the text opens with a review of basic concepts, advancing to surveys of random variables, the properties of expectation, conditional probability and expectation, and characteristic functions. Subsequent topics include infinite sequences of random variables, Markov chains, and an introduction to statistics. Complete solutions to some of the problems appear at the end of the book.

Basic Probability Theory

Measure, Integration, and Functional Analysis.

Measure, Integration, and Functional Analysis: 3 Measures on Topological Spaces4.4 Measures on Uncountably Infinite Product Spaces; 4.5 Weak Convergence of Measures; 4.6 References; Appendix on General Topology; A1 Introduction; A2 Convergence; A3 Product and Quotient Topologies; A4 Separation Properties and Other Ways of Classifying Topological Spaces; A5 Compactness; A6 Semicontinuous Functions; A7 The Stone-Weierstrass Theorem; A8 Topologies on Function Spaces; A9 Complete Metric Spaces and Category Theorems; A10 Uniform Spaces; Bibliography; Solutions to Problems; Subject Index

Robert B. (University of Illinois Ash, Urbana-Champaign U.S.A.)

Catherine A. Doleans-Dade

Catherine A. (University of Illinois Doleans-Dade, Urbana-Champaign U.S.A.)

Probability and Measure Theory, Second Edition, is a text for a graduate-level course in probability that includes essential background topics in analysis. It provides extensive coverage of conditional probability and expectation, strong laws of large numbers, martingale theory, the central limit theorem, ergodic theory, and Brownian motion. Clear, readable style Solutions to many problems presented in text Solutions manual for instructors Material new to the second edition on ergodic theory, Brownian motion, and convergence theorems used in statistics No knowledge of general topology required, just basic analysis and metric spaces Efficient organization

Probability and Measure Theory

Carol Ash

This book fills an educational void by adapting unique classroom-tested techniques that students find most congenial...that strip the shroud of mystery from an esoteric subject...that prepare students for applications of calculus in later courses.

The Calculus Tutoring Book

Developed by Claude Shannon and Norbert Wiener in the late Forties, information theory, or statistical communication theory, deals with the theoretical underpinnings of a wide range of communication devices: radio, television, radar, computers, telegraphy, and more. This book is an excellent introduction to the mathematics underlying the theory. Designed for upper-level undergraduates and first-year graduate students, the book treats three major areas: analysis of channel models and proof of coding theorems (Chapters 3, 7 and 8); study of specific coding systems (Chapters 2, 4, and 5); and study of statistical properties of information sources (Chapter 6). Among the topics covered are noiseless coding, the discrete memoryless channel, error correcting codes, information sources, channels with memory and continuous channels. The author has tried to keep the prerequisites to a minimum. However, students should have a knowledge of basic probability theory. Some measure and Hilbert space theory is helpful as well for the last two sections of Chapter 8, which treat time-continuous channels. An appendix summarizes the Hilbert space background and the results from the theory of stochastic processes necessary for these sections. The appendix is not self-contained, but will serve to pinpoint some of the specific equipment needed for the analysis of time-continuous channels. In addition to historic notes at the end of each chapter indicating the origin of some of the results, the author has also included 60 problems, with detailed solutions, making the book especially valuable for independent study.

Information Theory

Geared toward upper-level undergraduates and graduate students, this text surveys fundamental algebraic structures and maps between these structures. Its techniques are used in many areas of mathematics, with applications to physics, engineering, and computer science as well. Author Robert B. Ash, a Professor of Mathematics at the University of Illinois, focuses on intuitive thinking. He also conveys the intrinsic beauty of abstract algebra while keeping the proofs as brief and clear as possible. The early chapters provide students with background by investigating the basic properties of groups, rings, fields, and modules. Later chapters examine the relations between groups and sets, the fundamental theorem of Galois theory, and the results and methods of abstract algebra in terms of algebraic number theory, algebraic geometry, noncommutative algebra, and homological algebra, including categories and functors. An extensive supplement to the text delves much further into homological algebra than most introductory texts, offering applications-oriented results. Solutions to all problems appear in the text.

Robert B. Ash

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