Ph.D.  Comprehensive Examination Material, Probability  MATH 5319 

1.      Preliminaries

·        Axioms of Probability

·        Construction of Measures

·        Independence

·        Borel-Cantelli Lemmas

·        Product Spaces

·        Distributions and Expectations

2.      Weak Convergence

·        Characteristic Functions

·        Moment Generating Functions

·        Convergence in Distribution

3.      Independent Sums

·        Independence and Convolution

·        Weak Law of Large Numbers (WLLN)

·        Strong Law of Large Numbers (SLLN)

·        Central Limit Theorem (CLT)

·        Laws of the Iterated Logarithm (LIL)

4.      Dependent Random Variables

·        Conditional Probability

·        Conditional Expectation

·        Markov Chains

·        Stopping Times

5.      Martingales

·        Up-crossing inequality, Stopping Times

·        Martingale Convergence Theorem

·        Martingale Transforms

6.      Stationary Random Processes

·        Ergodic Theorems

·        Stationary Measures

·        Stationary Markov/Gaussian Processes

REFERENCES

Probability Essentials, Jean Jacod, Phillip Protter, Springer Verlag, 2004.

Probability Theory, S.R.S. Varadhan, AMS Courant Institute Lecture Notes, 2001.

Probability, Leo Breiman, Addison Wesley, 1968.  Paperback Classic(1992).