EECS 510: Probabilistic Techniques in Communication and Computation
Spring 2009


Lecture 1Introduction,Probability Review, Randomized Min-cut
Lecture 2Prob. Review cont'd., Basic inequalities: Jensen, Markov, Chebyshev Chernoff.
Lecture 3 Chernoff bounds - applications to Poisson trials and permutation routing.
Lecture 4 Weak and Strong Laws of large numbers, intro to stochastic processes and random walks.
Lecture 5Random walks and threshold crossings, log-moment generating functions, Wald's identity, applications to G/G/1 queues.
Lecture 6More of random walks and threshold crossings, conditional expectations, Martingales, Doob martingales, martingales and random graphs.
Lecture 7 Stopping times and stopped martingales, the martingale stopping theorem, examples of martingale stopping theorem- gambler's ruin, Wald's equation.
Lecture 8 Examples of martingale stopping theorem - Wald's identity for random walks; Azuma-Hoeffding Inequalities, applications to balls and bins.
Lecture 9 More on Azuma-Hoeffding, application to bounding the Chromatic number of random graphs, supper/sub-martingales.
Lecture 10Kolmogorov's inequalities, Martingale proof of the strong law of large numbers, the martingale convergence theorem, branching processes.
Lecture 11Markov Chains, basic definitions and examples (random walks of graphs and a randomized algorithm for 2-SAT).
Lecture 12Markov chains - state classification, expected hitting times.
Lecture 13 Strong Markov Property and first passage times.
Lecture 14 Ergodic classes, invariant distributions, finite Markov chains and Peron-Forbenius theory, Page Rank example.
Lecture 15 Markov chains with countable state spaces, convergence to steady-state, coupling, variation distance and mixing times.
Lecture 16More on coupling of Markov chains, applications to card shuffling. Introduction to renewal theory, Strong law for renewal processes, Elementary renewal theorem.
Lecture 17 Renewal theory and Markov chains, cover times, randomized algorithm for testing graph connectivity, key renewal theorem, Blackwell's theorem.
Lecture 18Renewal-reward processes, Residual life and age of renewal processes, Markov chains with rewards, Little's theorem.