Semester & Code: 2026 Fall MT4134
Course title: Probability Theory
Credits: 4
Prerequisites: Measure Theory and Integration
Introduction: This course gives an introduction to probability theory. The goal of this course is to introduce basic notions in probability and then move on to important topics such as Martingales and Markov chains. The topics covered in this course are essential for those interested in advanced probability theory, mathematical finance, mathematical biology, time series analysis, and related fields.
Course contents: Kolmogorov’s model of probability: The Probability Space, Events, properties of probability measures, independence of events. Random variable, distribution functions, decomposition of the distribution function, joint distributions, characteristic function. Bayes’ formula, Conditional expectation, and conditional probability, independence of σ-algebras and random variables. Regular conditional probability.
Borel-Cantelli lemmas, Kolmogorov 0-1 law, Central Limit Theorem of Lindeberg and Feller, various notions of convergence of a sequence of random variables.
Martingales, stopping time, Kolmogorov's, and Doob's inequalities. Almost sure and L2 convergence of martingale with bounded variance, Doob's up-crossing inequality, Submartingale convergence theorem.Doob's optional stopping theorem, Wald's first equation. Limit theorems: Weak/Strong law of large numbers.
References:
Probability Theory: Independence, Interchangeability, Martingales: Yuan Shih Chow and Henry Teicher (3e, 2003) Springer Texts in Statistics
Probability: A Graduate Course (Springer Texts in Statistics): A. Gut (2010) Springer
Probability Theory: R. G. Laha and V. K. Rohatgi (1979), Wiley
Probability Theory: K. B. Athreya and S.N. Lahiri (2006) Hindustan Book Agency TRIM/ 41
Introduction to Probability Models: S. M. Ross (2014) Academic Press
Introduction to the Theory of Probability and its Applications, Vol. 1: W. Feller (2008) Wiley
Introduction to Stochastic Processes: P. G. Hoel, S. C. Port and C.J. Stone (1986) Waveland Press Inc.
Lectures
03/08 The need for σ-algebra structure in the domain of definition of a measure, Lebesgue measure, and Lebesgue σ-algebra. An example of a real subset not in Lebesgue σ-algebra. Definition of an abstract measure, the definition of a probability measure. How to define a random variable mathematically? Explanation of Kolmogorov's model of random variables, Definition of measurable functions,
04/08 Notion of "almost sure". Construction of Bernoulli and Normal random variables. Note
Various notions of convergence of a sequence of random variables. Measure-theoretic interpretation of expectation. Integration of simple functions. Definition of integrable functions and their integration, convergence of a sequence of integrals, Notion of absolute continuity of measures, Radon-Nikodym Theorem, R-N derivative. Note. Statement of all major theorems on the convergence of integrals. Note
06/08 Definition of Borel σ-algebra, σ-algebra generated by a random variable. Independence of random numbers. The distribution measure of a random variable. Note
10/08 Given a sequence of events, the notion of the occurrence of infinitely many often. Revisit the definition of algebra, σ-algebra and monotone class. Definition of σ-algebra generated by a collection of subsets. A monotone algebra is a σ-algebra.
11/08 A monotone class generated by an algebra is a σ-algebra generated by that algebra (Monotone Class Theorem for sets). Note
13/08 Definition of π-class, λ-class generated by a collection of subsets. A collection that is a π-class, as well as a λ-class is a σ-algebra. The λ-class generated from a π-class is the σ-algebra generated by that π-class. Note
17/08 Kolmogorov's 0-1 Law, Statement and proof. Note
18/08 Statement of Borel Cantelli Theorem, Proof of Borel Cantelli Theorem. Note Consolidated Notes
Definition of Cumulative Distribution Function(CDF) and example; Properties of CDF. Definition of semi-algebra, extension of measure from a semi-algebra, The Lebesgue Stieltjes measure induced by a distribution function. Existence of a distribution measure for a given CDF; Existence of a random variable for a given CDF; Jump of CDF, countability of the set of jump points; Decomposition of a CDF as a convex combination of discrete and continuous CDFs. Note. Definition of Characteristic Function, Statement, and proof of the Lévy Inversion Formula. Levy's Continuity Theorem on the sequence of CDFs. Note
20/08 Example of the conditional expectation of the outcome of a rolling dice given a dependent random variable. Definition of the conditional expectation of a random variable(having finite expectation) given a σ-algebra, or given a measurable set. Definition of the conditional probability of an event given a measurable set. Proof of P(A|B)P(B)=P(A&B).
24/08 View conditional expectation as a Radon-Nikodym derivative. Note
25/08 Properties of the conditional expectation: linearity, monotonicity, MCT, FL, DCT. If X is G-measurable, E(XY|G)= XE(Y|G), and hence E(X|G)=X, with proofs. Note
27/08 Revisiting the definition of independence of random variables. Proof of the fact that the expectation of product of independent numbers (having finite expectation) is the product of expectations. Proof of E(X|G)=E(X) when X is independent of the sub σ algebra G with or without the previous theorem. Note Note
31/08 Tower property of Conditional expectation, Conditional Expectation as the projection of L^2 random variables in the subspace of random variables measurable w.r.t. a sub σ-algebra. Proof of V(X)>=V(E[X|G]), Note.
Reading assignment: Association inequality. Independence of X & Z does not imply "E[X|Y, Z]=E[X|Y]". An example to illustrate this. Statement: If σ(X) and D1 are independent to D2, then E[X|D1V D2]=E[X|D1 ]. Note
01/09 Quiz 1 out of 15
03/10 Properties of conditional probability, Definition of Regular conditional probability. Note.
07/09 Conditional expectation as integration wrt regular conditional probability measure. Definition of Conditional Distribution of X given a sub σ-algebra G.
08/09 Recollection of Carathéodory extension theorem. Note Definition of the n-dimensional distribution function.
10/09 Proof of Doob's theorem on the existence of regular conditional distribution measure of X given a sub σ-algebra G. Note
14/09 Holiday
15/09 Continuation
17/09 Quiz 2 out of 15
??/09 Midsem Exam out of 36
Mid Semester Break
05/10 Quiz Paper correction checking. Motivation for the Central Limit Theorem (CLT).
06/10 Lindeberg condition on a sequence of independent random variables. Examples and counterexamples.
08/10 Midsem correction checking. Statement, and proof of CLT of Lindeberg and Feller. Note
12/10 Definition of filtration, martingale, the example of SSRW. Stopping time relative to a filtration. [Notes]
13/10 Definition and a few properties of the stopping time σ-algebra. Closed martingales [Notes]
15/10 Decomposition of sub- and super-martingale, an example of a bounded martingale, a convex function of a martingale, with finite mean is a submartingale, [Notes]
19/10 Proof of Kolmogorov's inequality, and its extension.
20/10 Holiday
22/10 Various notions of convergence of a random sequence, Sufficient condition for convergence in probability 1a, 1b
26/10 Sufficient condition for almost sure convergence [Notes]
27/10 Martingale convergence theorem with bounded variance (in almost sure and L2 sense) [Note]
29/10 A lemma on up crossing and Proof of the lemma [Notes]
02/10 Proof of Doob's up-crossing inequality. Proof of submartingale convergence theorem [Notes]
03/11 Strong and Weak law of large number part 1
05/11 Strong and Weak Law of Large Numbers Part 2 Combined Note M-Z WLLN Note Recorded Lecture
09/11 Quiz 3 out of 15
10/11 Example where the mean of a martingale at a stopping time differs from the initial value. Doob's Optional Stopping Theorem (OST)
12/11 OST with stopping time having finite mean. Wald's Equation. Uniformly integrable martingale is closed and converges in L^1 Note
16/11 Discrete-time Markov chain and Chapman-Kolmogorov equation.
17/11 Showing corrections of Quiz 3, Example of a martingale that is not Markov. Note
19/11 Tutorial
Supplementary results:- 3, Supplementary results:- 4-5, Supplementary results:- 6-7
29/11 Endsem Test out of 34