**Courant Lecture Notes**

Volume: 7;
2001;
167 pp;
Softcover

MSC: Primary 60;

Print ISBN: 978-0-8218-2852-6

Product Code: CLN/7

List Price: $28.00

Individual Member Price: $22.40

**Electronic ISBN: 978-1-4704-1137-4
Product Code: CLN/7.E**

List Price: $28.00

Individual Member Price: $22.40

#### Supplemental Materials

# Probability Theory

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*S. R. S. Varadhan*

A co-publication of the AMS and the Courant Institute of Mathematical Sciences at New York University

S. R. S. Varadhan is recognized as a top expert in
probability theory. This volume presents topics in probability theory
covered during a first-year graduate course given by Varadhan at the
Courant Institute of Mathematical Sciences. The necessary background
material in measure theory is developed, including the standard
topics, such as extension theorem, construction of measures,
integration, product spaces, Radon-Nikodym theorem, and conditional
expectation.

In the first part of the book, characteristic functions are introduced,
followed by the study of weak convergence of probability distributions. Then
both the weak and strong limit theorems for sums of independent random
variables are proved, including the weak and strong laws of large numbers,
central limit theorems, laws of the iterated logarithm, and the Kolmogorov
three series theorem. The first part concludes with infinitely divisible
distributions and limit theorems for sums of uniformly infinitesimal
independent random variables.

The second part of the book mainly deals with dependent random variables,
particularly martingales and Markov chains. Topics include standard results
regarding discrete parameter martingales and Doob's inequalities. The standard
topics in Markov chains are treated, i.e., transience, and null and positive
recurrence. A varied collection of examples is given to demonstrate the
connection between martingales and Markov chains.

Additional topics covered in the book include stationary Gaussian processes,
ergodic theorems, dynamic programming, optimal stopping, and filtering. A large
number of examples and exercises is included. The book is a suitable text for a
first-year graduate course in probability.

S. R. S. Varadhan is the winner of the 2007 Abel Prize.
Varadhan was awarded the prize "for his fundamental contributions to
probability theory and in particular for creating a unified theory of large
deviations".
Read more here.

Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

#### Table of Contents

# Table of Contents

## Probability Theory

- Cover Cover11
- Other titles in this series i2
- General editors ii3
- Title page iii4
- Contents v6
- Preface vii8
- Measure theory 110
- Weak convergence 1928
- Independent sums 3544
- Dependent random variables 7382
- Martingales 109118
- Stationary stochastic processes 131140
- Dynamic programming and filtering 157166
- Bibliography 163172
- Index 165174
- Back Cover Back Cover1178

#### Readership

Graduate students and research mathematicians interested in probability theory and stochastic processes and in applications to economics, and finance.