Stochastic Models for Web 2.0

These are meant to be a rough set of notes and so could have some informal language. They give a quick reference to the background and material for the course but are not meant to be comprehensive. We will provide proofs for certain results but for most details we will refer the reader to the appropriate book or article. Notation: N = {1, 2, . . . } is the set of natural numbers, Z = {. . . ,−1, 0, 1, . . . } the set of whole numbers, R = (−∞,∞) is the real line, Z+ = {0}∪N is the set of non-negative whole numbers, and R+ = [0,∞) is the set of non-negative real numbers. Things of importance will be marked with ∗ with increasing number corresponding to increasing importance. Sources : Further discussion and details can be found in the following books: • D. Williams, “Probability with martingales,” Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1991. • J. R. Norris, “Markov chains,” Cambridge Series in Statistical and Probabilistic Mathematics, 2, Cambridge University Press, Cambridge, 2006. • P. Brémaud, “Markov chains: Gibbs fields, Monte Carlo simulation, and queues,” Texts in Applied Mathematics, 31, Springer-Verlag, New York, 1999. • S. N. Ethier and T. G. Kurtz, “Markov processes: Characterization and convergence,” Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. • P. Billingsley, “Probability and measure,” Third edition, Wiley Series in Probability and Mathematical Statistics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1995. • O. Kallenberg, “Foundations of modern probability,” Second edition, Probability and its Applications (New York), Springer-Verlag, New York, 2002.