Abhishek Halder

Review of preliminaries
     – Main objects: probability space , events , probability
     – Basic properties: countable additivity or -additivity, continuity of probability
     – Random variables, expectation and variance for discrete random variables, product spaces, pushforward of probability and change-of-variable formula
     – Important inequalities: Markov, Chebyshev, Hoeffding-Chernoff and its special case (Chernoff-Okamoto bound), why Hoeffding-Chernoff is stronger than CLT,
         Bennet's inequality and its relaxed case (Bernstein's inequality), Holder's inequality, Minkowski inequality
     – Poisson approximation of binomial, Lindebergh's CLT (important idea: probability as expected value of indicator)
Probability spaces
     – Definitions: algebra , -algebra, probability , smallest -algebra , semi-algebra , : smallest algebra that conatins
     – Construction of probability spaces , outer and inner measure and Caratheodory extension theorem, Monotone Class Theorem, approximation
         lemma, CDF, how to construct from CDF (use Heine-Borel Theorem), Borel -algebra and its extension to product spaces
Measurable maps and random variables
     – Inverse maps, measurable space, measurable functions and random variables, measurable random vectors
     – Induced probability measure: composition, measurability and continuity, measurability and limits
     – -algebra generated by maps
Integration and expectation
     – Simple functions and measurability
     – Definition of expectation and its properties (for simple functions), positive measurable functions or random variables
     – Monotone Convergence Theorem (MCT) for positive measurable functions, definition of expectation and MCT for general measurable functions, Fatou lemma,
         Dominated Convergence Theorem (DCT)
     – Expectation as integral, Lebesgue measure, distribution and density of a random variable, change-of-variables theorem, Riemann vs Lebesgue integral
     – Product -algebra and product probability measure, Fubini-Torelli theorem
Independence
     – Definition of independent events, -algebras and random variables
     – Dyadic expansion of uniform random variables, quantile transorm lemma
     – Groupings, Borel-Cantelli lemma and its applications: uniform continuity of integral, if $X_{n}$ iid exponential with then $mathcal{P}left(limsuplimits_{n to infty} frac{X_{n}}{log{n}} = 1right) = 1$ ,
         Borel zero-one law, Kolmogorov zero-one law: tail -algebra and applications, Savage-Hewitt zero-one law
Modes of Convergence
     – Sure, almost-sure, in probability and in distribution
     – $L_{p}$ convergence: uniform integrability, results on $L_{p}$ convergence
     – $L_{p}left(Omega, mathcal{B}, mathcal{P}right)$ is a complete normed vector space for , Riesez-Frechet theorem, absolute continuity of one measure w.r.t other, Radon-Nikodyn derivative
         theorem (von Neumann proof)
Convergence of random series
     – Truncation and equivalence, WLLN
     – Almost sure convergence of sum of independent (but not necessarily iid) random variables
     – Kolmogorov's inequality, SLLN, Kronecker's lemma
     – Kolmogorov's three series theorem
Conditional expectation and martingales
     – Conditional expectation: definition, examples, existence and uniqueness
     – Properties of conditional expectation: ten properties
     – Martingales: definition, submartingale and supermartingale, reverse martingale, examples
     – Martingale inequalities: Doob's inequality, Kolmogorv's second inequality, Doob's upcrossing inequality
     – Uniform integrability and martingales, sub and supermartingales, convergence theorems, reverse submartingale theorem
     – Applications of martingale convergence theorems: Levy's convergence theorem and its corollary, different proof of Kolmogorov zero-one law, probabilistic proof of
         the fact that continuous functions are dense in $L_{2}$ , Polya's urn scheme
     – Stopping times: definition of and $mathcal{B}_{tau}$ , properties, example: double down strategy of betting, random walk

Applied Probability (MATH 619)