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# UCD MAT 235A: Probability Theory

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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
*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