| Lecture 1 | Introduction,Probability Review, Randomized Min-cut
|
| Lecture 2 | Prob. 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 5 | Random walks and threshold crossings, log-moment
generating functions, Wald's identity, applications to G/G/1 queues. |
| Lecture 6 | More 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 10 | Kolmogorov's inequalities, Martingale proof
of the strong law of large numbers, the martingale
convergence theorem, branching processes. |
| Lecture 11 | Markov Chains, basic definitions and
examples (random walks of graphs and a randomized algorithm
for 2-SAT). |
| Lecture 12 | Markov 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 16 | More 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 18 | Renewal-reward processes, Residual life and
age of renewal processes, Markov chains
with rewards, Little's theorem. |