Integrated MSc-PhD Program


KSM4E13: Measure Theoretic Probability

Topics covered include  Probability space, Lebesgue measure,  Non-measurable sets, Random variables, Borel Probability measures on Euclidean  spaces, Examples of probability measures
on the line, A metric on the space of  probability measures on \mathbb{R}^d,  Compact subsets of \mathcal{P}(\mathbb{R}^d), absolute continuity and singularity, expectation, limit theorems for expectation, Lebesgue integral versus Riemann integral, Lebesgue spaces, some inequalities for expectations, change of variables, distribution of the sum and product, mean, variance, moments, independent random variables, product measures, independent sequences of random variables, some probability estimates, applications of first and second moment methods, weak law of large numbers, applications of weak law of large numbers, modes of convergence, uniform integrability, strong law of large numbers, Kolmogorov’s zero-one law, the law of iterated logarithm, Hoeffding’s inequality, random series with independent terms, Kolmogorov’s maximal inequality, central limit theorem (statement, heuristics, and discussion), central limit theorem (proof using characteristic functions), CLT for triangular arrays, limits of sums of random variables, Poisson convergence for rare events, Brownian motion, Brownian motion and Wiener measure, some continuity properties of Brownian paths (negative and positive results), and Lévy’s construction of Brownian motion.

Suggested texts:

          1. Rick Durrett, Probability: Theory and Examples.
          2. Patrick Billingsley, Probability and Measure, 3rd ed., Wiley India.
          3. Richard Dudley, Real Analysis and Probability, Cambridge University Press.
          4. Leo Breiman, Probability, SIAM: Society for Industrial and Applied Mathematics.