Lectures
05/08 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 probability measure. How to define a random variable mathematically? Explanation of Kolmogorov's model of random variables, Definition of measurable functions, Notion of "almost sure". Construction of Bernoulli and Normal random variable. 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 convergence of integrals. Note
6/08 Revisit 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.
8/08 A monotone class generated by an algebra is a σ-algebra generated by that algebra (Monotone Class Theorem for sets). Note
12/08 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
13/08 Definition of Borel σ-algebra, σ-algebra generated by a random variable. Independence of random numbers. The distribution measure of a random variable. Note Definition of Cumulative Distribution Function(CDF) and example; Properties of CDF.
19/08 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
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). View conditional expectation as a Radon-Nikodym derivative. Note
22/08 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
26/08 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. If X is independent of G, then E(X|G)= E(X). Note
27/08 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.
29/08 Association inequality Note. 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 ].
02/09 Quiz 1 out of 15 [Max=13.0, Min=1.0, Mean=9.4, Median=10.3, STD=2.9 ]
03/09 Tutorial. Properties of conditional probability, Definition of Regular conditional probability. Note
05/09 Conditional expectation as integration wrt regular conditional probability measure. Definition of Conditional Distribution of X given a sub σ-algebra G.
09/09 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 Given a sequence of events, the notion of occurrence of infinitely many often.
12/09 Statement of Borel Cantelli Theorem, Proof of Borel Cantelli Theorem. Note
17/09 Kolmogorov's 0-1 Law, Statement and proof. Proof of E(X|G)=E(X) when X is independent of the sub σ algebra G. Note
20/09 10.00 am to 12.00 pm Midsem Exam out of 36 [Max=36.0, Min=3.5, Mean=28.5, Median=32.0, STD=7.9 ]
Mid Semester Break
03/10 Definition of Characteristic Function, Statement, and proof of the Lévy Inversion Formula. Levy's Continuity Theorem on the sequence of CDFs. Note
07/10 Motivation for the Central Limit Theorem (CLT). Lindeberg condition on a sequence of independent random variables, Statement, and proof of CLT of Lindeberg and Feller. Note
08/10 Definition of filtration, martingale, the example of SSRW. Stopping time relative to a filtration. [Notes]
10/10 Definition and few properties of the stopping time σ-algebra. Closed martingales [Notes]
14/10 Decomposition of sub- and super-martingale, an example of a bounded martingale, a convex function of a martingale is a submartingale, [Notes]
15/10 Kolmogorov's inequality, and its extension.
17/10 Various notions of convergence of a random sequence, Sufficient condition for convergence in probability 1a, 1b
21/10 Sufficient condition for almost sure convergence [Notes]
22/10 Almost sure and L2 convergence of martingale with bounded variance [Note]
28/10 Strong and Weak law of large number part 1
29/10 Strong and Weak law of large number part 2 Combined Note M-Z WLLN Note Recorded Lecture
04/11 Quiz 2 out of 15 [Max=15,Min=0,Mean=11.7,Median=12.5,STD=3.6], Midsem correction checking.
05/11 Tutorial
07/11 A lemma on up crossing and Proof of the lemma [Notes]
11/11 Proof of Doob's up-crossing inequality. Proof of submartingale convergence theorem [Notes]
12/11 Example where the mean of a martingale at a stopping time differs from the initial value. Doob's Optional Stopping Theorem (OST)
14/11 OST with stopping time having finite mean. Wald's Equation. Uniformly integrable martingale is closed and converges in L^1 Note
18/11 Quiz 3 out of 15 [Max=15, Min=1.5, Mean=10.6, Median=11.3, STD=3.4]
19/11 Tutorial , Example of a martingale that is not Markov. Note / Supplementary results:- 3, Supplementary results:- 4-5, Supplementary results:- 6-7
27/11 Endsem Test 3 PM out of 34 [Max=33, Min=5, Mean=25.2, Median=27.5, STD=7.1]
The minimum cutoff for the grades are listed below.
30 D
40 C
50 C+
60 B
70 B+
80 A
90.5 A+