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! =

√

2π

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.