Description

The 2006 INFORMS Expository Writing Award-winning and best-selling author Sheldon Ross (University of Southern California) teams up with Erol Peköz (Boston University) to bring you this textbook for undergraduate and graduate students in statistics; mathematics; engineering; finance; and actuarial science.

This is a guided tour designed to give familiarity with advanced topics in probability without having to wade through the exhaustive coverage of the classic advanced probability theory books.

Topics include measure theory; limit theorems; bounding probabilities and expectations; coupling and Stein’s method; martingales; Markov chains; renewal theory; and Brownian motion. No other text covers all these advanced topics rigorously but at such an accessible level; all you need is calculus and material from a first undergraduate course in probability.

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• 1. Measure Theory and Laws of Large Numbers
Introduction
A Non-Measurable Event
Countable and Uncountable Sets
Probability Spaces
Random Variables
Expected Value
Almost Sure Convergence and the Dominated Convergence Theorem
Convergence in Probablitiy and in Distribution
Law of Large Numbers and Ergodic Theorem
Exercises

2. Stein's Method and Central Limit Theorems
Coupling
Poisson Approximation and Le Cam's Theorem
The Stein-Chen Method
Stein's Method for the Geometric Distribution
Stein's Method for the Normal Distribution

3. Conditional Expectation and Martingales
Conditional Expectation
Martingales
The Martingale Stopping Theorem
The Hoeffding-Azuma Inequality
Submartingales, Supermartingales, and a Convergence Theorem

4. Bounding Probabilities and Expectations
Jensen's Inequality
Probability Bounds via the Importance Sampling Identity
Chernoff Bounds
Second Moment and Conditional Expectation Inequalities
The Min-Max Identity and Bounds on the Maximum
Stochastic Orderings

5. Markov Chains
The Transition Matrix
The Strong Markov Property
Classification of States
Stationary and Limiting Distributions
Time Reversibility
A Mean Passage Time Bound

6. Renewal Theory
Some Limit Theorems of Renewal Theory
Renewal Reward Processes
6.3.1 Queueing Theory Applications of Renewal Reward Processes
Blackwell's Theorem
The Poisson Process

7. Brownian Motion
Continuous Time Martingales
Construction Brownian Motion
Embedding Variables in Brownian Motion
The Central Limit Theorem
Exercises.
• Citation

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