to the original problem about random walks, we see that
nS2n=0} = 2-2n-y-
2n
2n\
o
_
2 n
L ( 2 n ) 2 " e - 2 " v ^ _ y/2
n\n\ [Ln
n
e-
n
Vn]
2
Ly/n'
So the probability of being at the origin is about a constant times
n
-i/ 2 rpj^g -g
c o n s
i
s
t
e n
t ^ h
w n a
t We already know. We have seen
that the random walker tends to go a distance about a constant times
y/n. There are Cy/n such integer points, so it is very reasonable that
a particular one is chosen with probability a constant times n
- 1
/
2
.
We now consider the total number of times that the walker visits
the origin. Let Rn be the number of visits to the origin up through
time 2n. Then
Rn = Yo + Y\ + H- Yn,
where Yj = 1 if S2j = 0 and Yj = 0 if S2j -/=• 0. Note that E(Yj) =
P{S2j = 0}. Therefore,
n
E(Rn) = E(y0) + + E(yn) = £ P{5
2 j
= 0}
J= 0
^ V2 ,_1/2 2v^nV2
(Why is the last step true?) In particular, the expected number of
visits goes to infinity as n » oo. This indicates (and we will discuss
how to verify it in the next lecture) that the number of visits is infinite.
(There is a subtle point here. If
oo
R =
Rr^o

Yj
J
3 = 1
is the total number of visits to the origin, then we have demonstrated
that E(iZ) = oo. What we would like to conclude is that R oo.
There are positive finite random variables X with E(X) = oo (see
Problem 1-10), so in order to show that R = oo we need to show
more than E(i?) = oo.)
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