A Second Course in Probability – Sheldon M Ross, Erol A Peköz – 1st Edition


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
    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

    2. Stein's Method and Central Limit Theorems
    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
    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
  • Citation

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