COM-417: Advanced probability and applications

Lectures in this course (75)

Covers Mc Diarmid's inequality and its applications in probability and martingale theory.

Explores the proof and applications of the law of large numbers, emphasizing convergence of the empirical distribution.

Delves into the complexities of extending from finite variance to finite expectation assumptions.

Explores Kolmogorov's 0-1 law, showcasing cases of convergence and divergence in random variables based on the finiteness of expectations.

Explores the extension of the weak law of large numbers using St. Petersburg's paradox as an example.

Covers the concept of convergence in distribution for random variables and its significance in probability theory.

Explores the Curie-Weiss model, illustrating how temperature affects magnetization in materials.

Explores the central limit theorem and its equivalent criterion for convergence in distribution using continuous and bounded functions.

Explores the Central Limit Theorem, emphasizing the convergence towards a Gaussian distribution.

Explores the proof of the Central Limit Theorem through Lindeberg's principle and the convergence of random variables.

Explores an alternative proof of the Central Limit Theorem using characteristic functions to show the emergence of the Gaussian distribution.

Explores the coupon collector problem, analyzing the minimum number of balls needed for each bin to contain at least one ball.

Explores the properties of the Gumbel distribution and its application in the coupon collector problem with illustrative examples.

Explores moments in random variables and Carleman's theorem for uniquely determining a random variable by its moments.

Covers Hoeffding's inequality and concentration inequalities with a focus on sequences of random variables.

Covers Cramer's theorem and Hoeffding's inequality in the context of the large deviations principle.

Explains conditional expectation, conditioning events, probabilities calculation, and examples with dice and random variables.

Covers basic properties of conditional expectation and Jensen's inequality in probability theory.

Discusses a proposition on conditional expectation for two random variables, emphasizing independence and measurability.

Explores the definitions and properties of martingales in probability theory, including key concepts and examples.

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