Proposition 13.2.
If
Y
≥
0
, then for any
A
,
P
(
Y > A
)
≤
E
Y
A
.
Proof.
Let
B
=
{
Y > A
}
. Recall 1
B
is the random variable that is 1 if
ω
∈
B
and 0 otherwise. Note
1
B
≤
Y/A
. This is obvious if
ω /
∈
B
, while if
ω
∈
B
, then
Y
(
ω
)
/A >
1 = 1
B
(
ω
). We then have
P
(
Y > A
) =
P
(
B
) =
E
1
B
≤
E
(
Y/A
) =
E
Y
A
.
We now prove the WLLN.
Theorem 13.3.
Suppose the
X
i
are i.i.d. and
E

X
1

and
Var
X
1
are ﬁnite. Then for every
a >
0
,
P
±²
²
²
S
n
n

E
X
1
²
²
²
> a
³
→
0
as
n
→ ∞
.
Proof.
Recall
E
S
n
=
n
E
X
1
and by the independence, Var
S
n
=
n
Var
X
1
, so Var (
S
n
/n
) = Var
X
1
/n
. We
have
P
±²
²
²
S
n
n

E
X
1
²
²
²
> a
³
=
P
±²
²
²
S
n
n

E
±
S
n
n
³²
²
²
> a
³
=
P
±²
²
²
S
n
n

E
±
S
n
n
³²
²
²
2
> a
2
³
≤
E

S
n
n

E
(
S
n
n
)

2
a
2
=
Var (
S
n
n
)
a
2
=
Var
X
1
n
a
2
→
0
.
The inequality step follows from Proposition 13.2 with
A
=
a
2
and
Y
=

S
n
n

E
(
S
n
n
)

2
.
We now turn to the central limit theorem (CLT).
Theorem 13.4.
Suppose the
X
i
are i.i.d. Suppose
E
X
2
i
<
∞
. Let
μ
=
E
X
i
and
σ
2
= Var
X
i
. Then
P
±
a
≤
S
n

nμ
σ
√
n
≤
b
³
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 Spring '09
 wen
 Probability theory, Ω, Sn, Var Sn

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