**Student Mathematical Library**

Volume: 2;
1999;
99 pp;
Softcover

MSC: Primary 60;
Secondary 65

Print ISBN: 978-0-8218-2029-2

Product Code: STML/2

List Price: $21.00

Individual Price: $16.80

**Electronic ISBN: 978-1-4704-2123-6
Product Code: STML/2.E**

List Price: $21.00

Individual Price: $16.80

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# Lectures on Contemporary Probability

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*Gregory F. Lawler; Lester N. Coyle*

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

#### Table of Contents

# Table of Contents

## Lectures on Contemporary Probability

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- IAS Park City Mathematics Institute vii8 free
- Preface ix10 free
- Lecture 1. Simple Random Walk and Stirling's Formula 114 free
- Lecture 2. Simple Random Walk in Many Dimensions 922
- Lecture 3. Self-Avoiding Walk 1528
- Lecture 4. Brownian Motion 2134
- Lecture 5. Shuffling and Random Permutations 2740
- Lecture 6. Seven Shuffles are Enough (Sort of) 3346
- Lecture 7. Markov Chains on Finite Sets 3952
- Lecture 8. Markov Chain Monte Carlo 4760
- Lecture 9. Random Walks and Electrical Networks 5366
- Lecture 10. Uniform Spanning Trees 6376
- Lecture 11. Random Walk Simulations 6982
- Lecture 12. Other Simulations 7588
- Lecture 13. Simulations in Finance 8194
- Problems 8598
- Bibliography 99112
- Back Cover Back Cover1113

#### Readership

Advanced undergraduates, graduate students, and research mathematicians.

#### 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.

-- European Mathematical Society Newsletter

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