Limits and the Law of Large Numbers
Charles B. Moss
July 29, 2010
I. Almost Sure Convergence
A. Let
ω
represent the entire random sequence
{
Z
t
}
. As previously,
our interest typically centers around the averages of this sequence
b
n
(
ω
) =
1
n
n
t
=1
Z
t
(1)
C.
Definition 2.9
Let
{
b
n
(
ω
)
}
be a sequence of realvalued ran
dom variables. WE say that
b
n
(
ω
) converges
almost surely
to
b
,
written
b
n
(
ω
)
a.s.
−→
b
(2)
if and only if there exists a real number
b
such that
P
[
ω
:
b
n
(
ω
)
→
b
] = 1
.
(3)
1. The probability measure
P
describes the distribution of
ω
and determines the joint distribution function for the entire
sequence
{
Z
t
}
.
2. Other common terminology is that
b
n
(
ω
) converges to
b
with
probability 1 (w.p.1) or that
b
n
(
ω
) is strongly consistent for
b
.
3.
Example 2.10
: Let
¯
Z
n
=
q
n
n
i
=1
Z
t
(4)
1
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AEB 6571 Econometric Methods I
Professor Charles B. Moss
Lecture XV
Fall 2010
where
{
Z
t
}
is a sequence of independently and identically
distributed (i.i.d.)
random variables with E [
Z
t
] =
μ <
∞
.
Then
¯
Z
n
a.s.
−→
μ
(5)
by the Komolgorov strong law of large numbers (Theorem
3.1).
4.
Proposition 2.11
: Given
g
:
R
k
→
R
l
(
k, l <
∞
) and any
sequence
{
b
n
}
such that
b
n
a.s.
−→
b
(6)
wher
b
n
and
b
are
k
×
1 vectors, if
g
is continuous at
b
, then
g
(
b
n
)
a.s.
−→
g
(
b
)
.
(7)
5.
Theorem 2.12
: Suppose
a)
y
=
Xβ
0
+ ;
b)
X
n
a.s.
−→
0;
c)
X X
n
a.s.
−→
M
, is finite and positive definite.
Then
β
n
exists
a.s.
for all
n
suﬃciently large, and
β
n
a.s.
−→
β
0
.
d) Proof:
Since
X X/n
a.s.
−→
M
, it follows from Proposi
tion 2.11 that det (
X X/n
)
a.s.
−→
det (
M
).
Because
M
is positive definite by (c), det (
M
)
>
0.
It follows that
det (
X X/n
)
>
0
a.s.
for all
n
suﬃciently large, so (
X X
)
−
1
exists for all
n
suﬃciently large. Hence
ˆ
β
n
≡
X X
n
−
1
X y
n
(8)
exists for all
n
suﬃciently large. In addition,
ˆ
β
n
=
β
0
+
X X
n
−
1
X
n
(9)
It follows from Proposition 2.11 that
ˆ
β
n
a.s.
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 Spring '10
 Staff
 Probability theory, Professor Charles B. Moss

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