University of Washington Math 523A Lecture 4 Lecturer: Yuval Peres Summary: University of Washington Math 523A Lecture 4 Lecturer: Yuval Peres Friday, April 10, 2009 1 Problems Problem 1: In a sequence of fair coin tosses, find P(001 < 011), where w is the hitting time of the word w. For example, in the sequence 010111001 . . ., 011 = 5 and 001 = 9, so the complement of the above event occurs. Note that although the probability of any given sequence of length 3 occurring at a particular location is 1/8, the probability that one sequence occurs before another is not necessarily 1/2, but depends on the particular pair of sequences. 2 Basic results on hitting times for SRW Last time we used the Optional Stopping Theorem to show that for a simple random walk {St} on Z, we have Pk[n < 0] = k/n and Ek{0,n} = k(n - k). Transforming [0, n] to a general interval, if a, b, x Z with x [a, b], then Ex{a,b} = (x - a)(b - x). (2.1) For example, E0{-n,n} = n2 . Thus, for a simple random walk {Yt} on [0, ), we have Collections: Computer Technologies and Information Sciences