 
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, Sk1 = 0, Xk = 1)  P0(X0 = 1, Sk1 = 0, Xk = 1, 1 < k)
=
1
4
P(Sk1 = 0S1 = 1) 
1
4
P(1 < k, Sk1 = 0S1 = 1)
In order to compute the later probability, we consider the reflected mapping:
: (X1, X2, ..., Xk1)  (X1, X2, ..., X1 , X1+1, ..., Xk1 := (Y1, ..., Yk1)
Note is a bijection and is identity if 1 k.
map the event {X1 = 1, 1 < k, Sk1 = 0} to {Y1 = 1, 1 < k, Sk1 = 2}
