Stopping Times: Martingales and Brownian Motion

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Explores transformations of Brownian motion, Markov properties, and scaling invariance.

Explores the proof of the martingale convergence theorem and the concept of stopping time in square-integrable martingales.

Explores sub- and supermartingales, stopping times, and their applications in stochastic processes.

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Discusses convergence, martingales, Brownian motion, joint laws, testing procedures, and stop times.

Explores the generalization of martingales to sub- and supermartingales with a focus on convergence properties.

Covers the concept of Doob's martingale and its properties, including integrability and convergence theorem.

Covers the optional stopping theorem for martingales, providing a detailed proof and discussing its implications.

Explores three stopping theorems in martingales and Brownian motion.

Explores spectral gap and mixing time in Markov chains, including their definitions and behavior.

Explores ergodicity and stationary distribution in Markov chains, emphasizing convergence properties and unique distributions.

Covers the proof of the martingale convergence theorem and the convergence of the martingale sequence almost surely.

Explores martingales, adaptability, and stopping times in stochastic processes.

Covers Markov chains, focusing on reversibility, convergence, ergodicity, and applications.

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Explores stopping times, the optional stopping theorem, F-measurable random variables, and martingales.

Explores invariant distributions, recurrent states, and convergence in Markov chains, including practical applications like PageRank in Google.

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