1.6. Exercises 45
Exercise 1.18. Suppose that Sn is a biased one-dimensional random
walk. To be more specific, let p 1/2 and
Sn = X1 + · · · + Xn,
where X1,...,Xn are independent with
P{Xj = 1} = 1 P{Xj = −1} = p.
Show that there is a ρ 1 such that as n ∞,
P{S2n = 0}
ρn
1

πn
.
Find ρ explicitly. Use this to show that with probability one the
random walk does not return to the origin infinitely often.
Exercise 1.19. Suppose δn is a sequence of real numbers with |δn|
1 and such that

j=1
|δn| ∞.
Let
sn =
n
j=1
(1 + δj).
Show that the limit s∞ = limn→∞ sn exists and is strictly positive.
Moreover, there exists an N such that for all n N,
1
sn
s∞
2

j=n+1
|δj|.
Exercise 1.20. Find the number t such that
n! =


nn+
1
2
e−n
1 +
t
n
+
O(n−2)
.
Exercise 1.21. Prove that

−∞
e−x2/2
dx =

2π.
Hint: There are many ways to do this but direct antidifferentiation
is not one of them. One approach is to consider the square of the
left-hand side; write it as a double (iterated) integral, and then use
polar coordinates.
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