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Softcover ISBN:  9780821828526 
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Softcover ISBN:  9780821828526 
Product Code:  CLN/7 
List Price:  $30.00 
MAA Member Price:  $27.00 
AMS Member Price:  $24.00 
eBook ISBN:  9781470411374 
Product Code:  CLN/7.E 
List Price:  $28.00 
MAA Member Price:  $25.20 
AMS Member Price:  $22.40 
Softcover ISBN:  9780821828526 
eBook ISBN:  9781470411374 
Product Code:  CLN/7.B 
List Price:  $58.00 $44.00 
MAA Member Price:  $52.20 $39.60 
AMS Member Price:  $46.40 $35.20 

Book DetailsCourant Lecture NotesVolume: 7; 2001; 167 ppMSC: Primary 60;
S. R. S. Varadhan is recognized as a top expert in probability theory. This volume presents topics in probability theory covered during a firstyear 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, RadonNikodym 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 firstyear 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.ReadershipGraduate students and research mathematicians interested in probability theory and stochastic processes and in applications to economics, and finance.

Table of Contents

Chapters

Chapter 1. Measure theory

Chapter 2. Weak convergence

Chapter 3. Independent sums

Chapter 4. Dependent random variables

Chapter 5. Martingales

Chapter 6. Stationary stochastic processes

Chapter 7. Dynamic programming and filtering


Additional Material

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S. R. S. Varadhan is recognized as a top expert in probability theory. This volume presents topics in probability theory covered during a firstyear 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, RadonNikodym 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 firstyear 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.
Graduate students and research mathematicians interested in probability theory and stochastic processes and in applications to economics, and finance.

Chapters

Chapter 1. Measure theory

Chapter 2. Weak convergence

Chapter 3. Independent sums

Chapter 4. Dependent random variables

Chapter 5. Martingales

Chapter 6. Stationary stochastic processes

Chapter 7. Dynamic programming and filtering