This is a graduate level class on probability theory, geared towards students who are interested in a rigorous development of the subject. It is likely to be useful for students specializing in communications, networks, signal processing, stochastic control, machine learning, and related areas. In general, the course is not so much about computing probabilities, expectations, densities etc. Instead, we will focus on the 'nuts and bolts' of probability theory, and aim to develop a more intricate understanding of the subject. For example, emphasis will be placed on deriving and proving fundamental results, starting from the basic axioms.

Course Contents:

Probability Spaces, σ-algebras, events, probability measures

Borel Sets and Lebesgue measure

Conditioning, Bayes' rule

Independence

Borel-Cantelli Lemmas

Measurable functions, random variables

Distribution functions, types of random variables

Joint distributions, transformation of random variables

Convergence of random variables, various notions of convergence

Central limit theorem

The laws of large numbers (the weak and strong laws)

Module no.

Module Name

Lecture Names

0

Preliminaries

Set Theory

Real Analysis basics

Cardinality and Countability

1

Probability Measures

Probability Spaces

Properties of Probability Measures

Discrete Probability Spaces

Borel Sets and Lebesgue Measure

Infinite Coin Toss Model

Conditional Probability and Independence

Borel-Cantelli Lemmas

2

Random Variables

RVs as measurable functions

Probability law, types of RVs, and CDF

Multiple Random Variables and Independence

Jointly Continuous Random Variables,

Conditional Distributions

Sums of Random Variables

General Transformations of Random Variables, Jacobian formula

3

Integration and Expectation

Abstract Integration

Properties of Abstract Integrals

Monotone Convergence Theorem

Integration over Different Spaces

Integration of Continuous Random Variables,

Radon-Nikodym theorem

Fatou's Lemma and Dominated Convergence Theorem

Variance and Covariance

Conditional Expectation and MMSE estimate

4

Transforms

Probability Generating Functions

Moment Generating Functions

Characteristic Functions

Inversion Theorem and Uniqueness of the Inversion

Concentration Inequalities

5

Limit theorems

Convergence of Random Variables and related theorems

Weak Law of Large Numbers

Strong Law of Large Numbers

Central limit theorem, Multi-variate Gaussian Distribution

There will be no official pre-requisites. Although
the course will build up from the basics, it will be
taught at a fairly sophisticated level. Familiarity
with concepts from real analysis will also be
useful. Perhaps the most important prerequisite for
this class is an intangible one, namely
mathematical maturity.

Probability and Random Processes by Geoffrey R. Grimmett and David R. Stirzaker. Oxford University Press, 3rd edition, 2001.

Probability with Martigales by D.
Williams, Cambridge University Press,
1991.

A First Look at Rigorous Probability Theory by J. Rosenthal, World Scientific Publishing Co Pte Ltd; 2nd Revised edition, 2006.

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