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).