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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


Source: Anderson, Richard - Department of Computer Science and Engineering, University of Washington at Seattle


Collections: Computer Technologies and Information Sciences