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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


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


Collections: Computer Technologies and Information Sciences