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Mathematics LibreTexts

UCD MAT 235A: Probability Theory

  • Page ID
    1098
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    1. Measure Theory

    1. Probability Spaces
    2. Distributions
    3. Random Variables
    4. Integration
    5. Properties of the Integral 6. Expected Value

    2. Laws of Large Numbers

    1. Independence
    2. Weak Laws of Large Numbers
    3. Borel-Cantelli Lemmas
    4. Strong Law of Large Numbers
    *5. Convergence of Random Series
    *6. Large Deviations

    3. Central Limit Theorems

    1. The De Moivre-Laplace Theorem
    2. Weak Convergence
    3. Characteristic Functions
    4. Central Limit Theorems
    *5. Local Limit Theorems
    6. Poisson Convergence
    *7. Stable Laws
    *8. Infinitely Divisible Distributions
    *9. Limit Theorems in Rd

    4. Random Walks

    1. Stopping Times
    2. Recurrence
    *3. Visits to 0, Arcsine Laws
    *4. Renewal Theory

    5. Martingales

    1. Conditional Expectation
    2. Martingales, Almost Sure Convergence
    3. Examples
    4. Doob's Inequality, Lp Convergence
    5. Uniform Integrability, Convergence in L1
    6. Backwards Martingales
    7. Optional Stopping Theorems

    6. Markov Chains

    1. Definitions
    2. Examples
    3. Extensions of the Markov Property
    4. Recurrence and Transience
    5. Stationary Measures
    6. Asymptotic Behavior
    *7. Periodicity, Tail sigma-field
    *8. General State Space

    7. Ergodic Theorems

    1. Definitions and Examples
    2. Birkhoff's Ergodic Theorem
    3. Recurrence
    *4. A Subadditive Ergodic Theorem
    *5. Applications

    8. Brownian Motion

    1. Definition and Construction
    2. Markov Property, Blumenthal's 0-1 Law
    3. Stopping Times, Strong Markov Property
    4. Maxima and Zeros
    5. Martingales
    6. Donsker's Theorem
    *7. Empirical Distributions, Brownian Bridge
    *8. Laws of the Iterated Logarithm

    Appendix: Measure Theory

    1. Caratheodary's Extension Theorem
    2. Which sets are measurable?
    3. Kolmogorov's Extension Theorem
    4. Radon-Nikodym Theorem
    5. Differentiating Under the Integral