Applied Probability (MATH 619)
by Dr. Dmitry Panchenko
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 changeofvariable formula
– Important inequalities: Markov, Chebyshev, HoeffdingChernoff and its special case (ChernoffOkamoto bound), why HoeffdingChernoff 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 , semialgebra , : 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 HeineBorel 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, changeofvariables theorem, Riemann vs Lebesgue integral
– Product algebra and product probability measure, FubiniTorelli theorem
Independence
– Definition of independent events, algebras and random variables
– Dyadic expansion of uniform random variables, quantile transorm lemma
– Groupings, BorelCantelli lemma and its applications: uniform continuity of integral, if iid exponential with then , Borel zeroone law, Kolmogorov zeroone law: tail algebra and applications, SavageHewitt zeroone law
Modes of Convergence
– Sure, almostsure, in probability and in distribution
– convergence: uniform integrability, results on convergence
– is a complete normed vector space for , RiesezFrechet theorem, absolute continuity of one measure w.r.t other, RadonNikodyn 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 zeroone law, probabilistic proof of
the fact that continuous functions are dense in , Polya's urn scheme
– Stopping times: definition of and , properties, example: double down strategy of betting, random walk
