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

π
−π
φ(t)n
dt =
1


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 =


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
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