EECS 510: Probabilistic Techniques in Communication and Computation

Spring 2009

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. |