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.)