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University of Washington Math 523A Lectures 10 and 11
 

Summary: University of Washington Math 523A
Lectures 10 and 11
Lecturer: Yuval Peres
May 1 and 4, 2009
1 reflection principle
we wish to compute the distribution of 1 where the simple random walk start from
zero.
note that we have:
P0(1 = k) = P0(X0 = -1, Sk-1 = 0, Xk = 1) - P0(X0 = -1, Sk-1 = 0, Xk = 1, 1 < k)
=
1
4
P(Sk-1 = 0|S1 = -1) -
1
4
P(1 < k, Sk-1 = 0|S1 = -1)
In order to compute the later probability, we consider the reflected mapping:
: (X1, X2, ..., Xk-1) - (X1, X2, ..., X1 , -X1+1, ..., -Xk-1 := (Y1, ..., Yk-1)
Note is a bijection and is identity if 1 k.
map the event {X1 = -1, 1 < k, Sk-1 = 0} to {Y1 = -1, 1 < k, Sk-1 = 2}

  

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

 

Collections: Computer Technologies and Information Sciences