Probability Theory

i) Revision of Measure theory: probability spaces, distributions, random vari ables, standard random variable examples, expected value, inequalities (Holder, Cauchy-Schwarz, Jensen, Markov, Chebyshev) , convergence notions(convergence in probability and almost sure, Lp), application of DCT, MCT, Fatou with ex amples, Revision of Fubinis theorem.
ii) Independence, sum of random variables, constructing independent random vari ables, weak law of large numbers, Borel-Cantelli lemmas, First and Second Moment methods, Chernoff bounds and some applications.
iii) Strong law of large number, Kolmogorov 0 - 1 law. Convergence of random series. Kolmogorovs three series theorem.
iv) Weak convergence, tightness, characteristic functions with examples, Central limit theorem (iid sequence and triangular array).

Suggested Texts :
(a) Rick Durrett: Probability (Theory and Examples).
(b) Patrick Billingsley: Probability and measure.
(c) Robert Ash: Basic probability theory.
(d) Leo Breiman: Probability.
(e) David Williams: Probability with Martingales.

https://www.isibang.ac.in/~adean/infsys/database/MMath/C12PT.html