1.6. Exercises 41

Hint: Write

log n +

1

2

− log

1

2

=

n+ 1

2

1

2

1

x

dx,

and estimate

1

j

−

j+ 1

2

j− 1

2

dx

x

.

Exercise 1.3. Show that there is a c 0 such that the following is

true. For every real number r and every integer n,

(1.32)

e−cr2/n

≤

er

1 −

r

n

n

≤

ecr2/n.

Exercise 1.4. Find constants a1,a2 such that the following is true

as n → ∞,

1 −

1

n

n

=

e−1

1 +

a1

n

+

a2

n2

+ O

(

n−3

)

.

Exercise 1.5. Let Sn be a one-dimensional simple random walk and

let

pn = P{S2n = 0 | S0 = 0}.

• Show that

(1.33) pn+1 = pn

2n + 1

2n + 2

,

and hence

pn =

1 · 3 · 5 · · · (2n − 1)

2 · 4 · 6 · · · (2n)

.

• Use the relation (1.33) to give another proof that there is a

c such that, as n → ∞,

pn ∼

c

√

n

.

(Our work in this chapter shows, in fact, that c = 1/

√

π, but you do

not need to prove this here.)

Exercise 1.6.

• Show that if X is a nonnegative random variable, then

lim

n→∞

E[X 1{X ≤ n}] = lim

n→∞

E[X ∧ n] = E[X].