Syllabus:
� Sigma-algebras, axioms of probability, pi - lamda theorem (proof can
be skipped),
uniqueness of extension for probability measures. Examples of countable
probability spaces, Borel sigma-algebra on the real line and standard
probability
distributions on the real line.
Construction of Lebesgue measure (statement alone). Random variables
and
examples. Push-forward of a probability measure (sketch of proof) .
Borel
probability measures on Euclidean spaces as push-forward of Lebesgue
measure (statement alone); Cumulative distribution function and
properties.
� General definition of expectation and properties. Change of variables. Review
of conditional distribution and conditional expectation, General definition, Examples.
� Limit theorems: Monotone Convergence Theorem (MCT) (without proof),
Fatous Lemma, Dominated Convergence Theorem (DCT), Bounded Convergence
Theorem (BCT), Cauchy-Schwartz, Jensen and Chebyshev inequalities.
� Different modes of convergence and their relations, Weak Law of large
numbers, First and Second Borel-Cantelli Lemmas, Strong Law of large
numbers
(proof under finite variance).
� Characteristic functions, properties, Inversion formula and Levy
continuity theorem (statements only), CLT in i.i.d. finite variance
case. Slutsky�s Theorem.
� Introduction to Finite Markov chains - Definition. Random mapping
representation. Examples. Irreducibility and aperiodicity. Stationary
distribution and
reversibility. Random walks on graphs.
Reference Texts:
(a) N. Lanchier: Stochastic Modelling.
(b) W. Feller: Introduction to Probability: Theory and Applications - Vol. I and II..
(c) J. Pitman: Probability.
(d) Sheldon Ross: Probability Models.
(e) Santosh S. Venkatesh: Theory of Probability - Explorations and Applications.
(f) R. Meester: A Natural Introduction to Probability Theory.
(g) S. R. Athreya and V. S. Sunder: Measure and Probability
https://www.isibang.ac.in/~adean/infsys/database/Bmath/PT3.html