Probility Theory

by Varadhan, S. R. S.

• ISBN13: 9780821828526
• ISBN10: 0821828525
• Format: Paperback
• Copyright: 2001-09-01
• Publisher: Amer Mathematical Society

Summary

This volume presents topics in probability theory covered during a first-year graduate course given 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.

Table of Contents

Preface	vii
Measure Theory	1	(18)
Introduction	1	(2)
Construction of Measures	3	(4)
Integration	7	(6)
Transformations	13	(1)
Product Spaces	14	(2)
Distributions and Expectations	16	(3)
Weak Convergence	19	(16)
Characteristic Functions	19	(3)
Moment-Generating Functions	22	(2)
Weak Convergence	24	(11)
Independent Sums	35	(38)
Independence and Convolution	35	(2)
Weak Law of Large Numbers	37	(3)
Strong Limit Theorems	40	(3)
Series of Independent Random Variables	43	(5)
Strong Law of Large Numbers	48	(1)
Central Limit Theorem	49	(5)
Accompanying Laws	54	(5)
Infinitely Divisible Distributions	59	(7)
Laws of the Iterated Logarithm	66	(7)
Dependent Random Variables	73	(36)
Conditioning	73	(6)
Conditional Expectation	79	(2)
Conditional Probability	81	(3)
Markov Chains	84	(5)
Stopping Times and Renewal Times	89	(1)
Countable State Space	90	(8)
Some Examples	98	(11)
Martingales	109	(22)
Definitions and Properties	109	(3)
Martingale Convergence Theorems	112	(3)
Doob Decomposition Theorem	115	(2)
Stopping Times	117	(3)
Up-crossing Inequality	120	(1)
Martingale Transforms, Option Pricing	121	(2)
Martingales and Markov Chains	123	(8)
Stationary Stochastic Processes	131	(26)
Ergodic Theorems	131	(4)
Structure of Stationary Measures	135	(2)
Stationary Markov Processes	137	(4)
Mixing Properties of Markov Processes	141	(2)
Central Limit Theorem for Martingales	143	(4)
Stationary Gaussian Processes	147	(10)
Dynamic Programming and Filtering	157	(6)
Optimal Control	157	(1)
Optimal Stopping	158	(3)
Filtering	161	(2)
Bibliography	163	(2)
Index	165