### MA7040 Advanced Probability Theory

#### Course Details

**Description:**

This course aims to introduce advanced topics in probability theory which will equip graduate students with tools to start research in this area.

**CourseContent:**

Independent Sums: Weak/Strong Law of Large Numbers, Law of Iterated Logarithm.

Central Limit Theorem: Berry-Essen and Lindeberg Theorems, Some Extensions. Convergence of Measures: Modes of Convergence, Infinite divisibility, Invariance Principle.

Wiener's Measure: Gaussian and Markov Aspects

Conditioning and Martingales: Conditional Expectation, Discrete and Continuous Martingales, Doob's Inequality and stopping time theorem, Diffusions.

#### Course References:

TextBooks:

Stroock, D. W. Probability theory. An analytic view. Second edition. Cambridge University Press, Cambridge, 2011.

ReferenceBooks:

1. Varadhan, S. R. S. Probability theory. Courant Lecture Notes in Mathematics, 7. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001.

2.Dudley, R. M. Real analysis and probability. Revised reprint of the 1989 original. Cambridge Studies in Advanced Mathematics, 74. Cambridge University Press, Cambridge, 2002.

3. Billingsley, P. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.

4. Billingsley, P. Probability and measure. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012.

Prerequisite:MA5400 or Equivalent