1.1. Simple random walk 9
This gives the value C0 =

(see Exercise 1.21), and hence Stir-
ling’s formula can be written as
n! =


nn+
1
2
e−n
1 +
O(n−1)
.
The limit in (1.7) is a statement of the central limit theorem (CLT)
for the random walk,
lim
n→∞
P{a

2n S2n b

2n} =
b
a
1


e−x2/2
dx.
1.1.4. Returns to the origin.

Recall that the sum

n=1
n−a
converges if a 1 and diverges otherwise.
We now consider the number of times that the random walker
returns to the origin. Let Jn = 1{Sn = 0}. Here we use the indicator
function notation: if E is an event, then 1E or 1(E) is the random
variable that takes the value 1 if the event occurs and 0 if it does not
occur. The total number of visits to the origin by the random walker
is
V =

n=0
J2n.
Note that
E[V ] =

n=0
E[J2n] =

n=0
P{S2n = 0}.
We know that P{S2n = 0} c/

n as n ∞. Therefore,
E[V ] = ∞.
It is possible, however, for a random variable to be finite yet have an
infinite expectation, so we need to do more work to prove that V is
actually infinite.

A well-known random variable with infinite expectation is that obtained
from the St. Petersburg Paradox. Suppose you play a game where you flip a
coin until you get tails. If you get k heads before flipping tails, then your payoff
is
2k.
The probability that you get exactly k heads is the probability of getting
Previous Page Next Page