University of Washington Math 523A Lecture 6 Lecturer: Yuval Peres Summary: University of Washington Math 523A Lecture 6 Lecturer: Yuval Peres Friday, April 17, 2009 1 Expected hitting times for words 1.1 Review of Li's martingale method Setting: X1, X2, . . . are IID and take values in some finite alphabet A, with P(Xi = a) = pa a A. Given a word w Ak , the hitting time of w is w = min{t 1 : (Xt-k+1, . . . , Xt) = w}. (Note that we must in fact have w k if the sequence starts at time 1. In the next section we consider a generalized setting in which the sequence may start before time 1 and hence w can be less than k.) Last time we computed Ew using the following martingale scheme. Martingale method (Li 1980) Think of a gambler, Glinda, making a sequence of fair bets on w coming up, starting at time t. First Glinda bets on the 1st digit of w, then continues to bet on each successive digit in a way that makes all the bets fair (i.e. her expected winnings are 0 at each step): · At time t, Glinda puts a dollar on Xt = w1. She gets 0 if wrong, gets 1 pw1 Collections: Computer Technologies and Information Sciences