1.6. Exercises 43

distribution of Xj is symmetric about the origin, Var[Xj] = E[Xj 2] =

σ2, E[|Xj|3] ∞. Also assume,

P{Xj = 0} 0, P{Xj = 1} 0.

The goal of this exercise is to prove

lim

n→∞

√

2πσ2n P{Sn = 0} = 1.

Prove the following facts:

• The characteristic function of X1 + · · · + Xn is

φn.

• For every 0 ≤ π there is a ρ 1 such that |φ(t)| ρ for

≤ |t| ≤ π.

•

P{Sn = 0} =

1

2π

π

−π

φ(t)n

dt =

1

2π

√

n

π

√

n

−π

√

n

φ(t/

√

n)n

dt.

• There is a c such that for |t| ≤ π,

φ(t) − 1 −

σ2 t2

2

≤ c

t3.

•

lim

n→∞

π

√

n

−π

√

n

φ(t/

√

n)n

dt =

∞

−∞

e−σ2t2/2

dt =

√

2π

σ

.

Hint: You will probably want to use (1.32).

Exercise 1.11. Suppose A is a finite subset of

Zd

and

F : ∂A → R, g : A → R

are given functions. Show that there is a unique extension of F to A

such that

LF (x) = −g(x), x ∈ A.

Give a formula for F .

Exercise 1.12. Suppose A is a finite subset of Zd and

F : ∂A → R, f : A → R