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Lectures on Contemporary Probability
Gregory F. Lawler, Duke University, Durham, NC, and Lester N. Coyle, Loyola College, Baltimore, MD
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Student Mathematical Library
1999; 99 pp; softcover
Volume: 2
ISBN-10: 0-8218-2029-X
ISBN-13: 978-0-8218-2029-2
List Price: US$21 Institutional Members: US$16.80
All Individuals: US\$16.80
Order Code: STML/2

Random Walk and the Heat Equation - Gregory F Lawler

This volume is based on classes in probability for advanced undergraduates held at the IAS/Park City Mathematics Institute (Utah). It is derived from both lectures (Chapters 1-10) and computer simulations (Chapters 11-13) that were held during the program. The material is coordinated so that some of the major computer simulations relate to topics covered in the first ten chapters. The goal is to present topics that are accessible to advanced undergraduates, yet are areas of current research in probability. The combination of the lucid yet informal style of the lectures and the hands-on nature of the simulations allows readers to become familiar with some interesting and active areas of probability.

The first four chapters discuss random walks and the continuous limit of random walks: Brownian motion. Chapters 5 and 6 consider the fascinating mathematics of card shuffles, including the notions of random walks on a symmetric group and the general idea of random permutations.

Chapters 7 and 8 discuss Markov chains, beginning with a standard introduction to the theory. Chapter 8 addresses the recent important application of Markov chains to simulations of random systems on large finite sets: Markov Chain Monte Carlo.

Random walks and electrical networks are covered in Chapter 9. Uniform spanning trees, as connected to probability and random walks, are treated in Chapter 10.

The final three chapters of the book present simulations. Chapter 11 discusses simulations for random walks. Chapter 12 covers simulation topics such as sampling from continuous distributions, random permutations, and estimating the number of matrices with certain conditions using Markov Chain Monte Carlo. Chapter 13 presents simulations of stochastic differential equations for applications in finance. (The simulations do not require one particular piece of software. They can be done in symbolic computation packages or via programming languages such as C.)

The volume concludes with a number of problems ranging from routine to very difficult. Of particular note are problems that are typical of simulation problems given to students by the authors when teaching undergraduate probability.

This book is published in cooperation with IAS/Park City Mathematics Institute.

Reviews

"Well-written booklet ... The authors ... present topics that are accessible to advanced undergraduates and ... show the appeal of parts of modern probability, and make the lectures very attractive."

"This nice, short monograph contains material from lectures and computer labs held at the IAS/Park City Mathematics Institute. The lectures present areas of modern probability theory that are current areas of research. The material is accessible to undergraduate students with a modest background in probability. This collection of lectures will be an excellent supplement to any intermediate probability course."

-- Journal of the American Statistical Association

"In less than a hundred pages, Lawler and Coyle set out what a student should really know after a course in probability theory learned from a text maybe four times the length, but written in a style students should find accessible before they take any such course. Almost from the start, the authors describe the unsolved problems that fire current research both to inspire the undergraduate and to clarify the current shape of the theory. Highly recommended."

-- CHOICE

"It is a beautiful ... book of high pedagogical value, easy to read, and focusing on the ideas rather than mathematical rigor of completeness."

-- Zentralblatt MATH

• Simple random walk and Stirling's formula
• Simple ramdon walk in many dimensions
• Self-avoiding walk
• Brownian motion
• Shuffling and random permutations
• Seven shuffles are enough (sort of)
• Markov chains on finite sets
• Markov chain Monte Carlo
• Random walks and electrical networks
• Uniform spanning trees
• Random walk simulations
• Other simulations
• Simulations in finance
• Problems
• Bibliography