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Limits and the Law of Large Numbers
Lecture XV
I.
Almost Sure Convergence
A.
White, Halbert.
Asymptotic Theory for Econometricians
(New York: Academic
Press, 1984). Chapter II.
B.
Let
represent the entire random sequence
t
Z
.
As discussed last time, our
interest typically centers around the averages of this sequence:
n
t
t
n
Z
n
b
1
1
C.
Definition 2.9: Let
n
b
be a sequence of realvalued random variables. We
say that
n
b
converges
almost surely
to
b
, written
b
b
s
a
n
.
.
if and only if there exists a real number
b
such that
1
:
b
b
P
n
.
1.
The probability measure
P
describes the distribution of
and determines
the joint distribution function for the entire sequence
t
Z
.
2.
Other common terminology is that
n
b
converges to
b
with probability
1
(w.p.1) or that
n
b
is strongly
consistent for
b
.
3.
Example 2.10: Let
n
t
t
n
Z
n
Z
1
1
where
t
Z
is a sequence of independently and identically distributed
(i.i.d.) random variables with
t
EZ
.
Then
.
.
s
a
n
Z
by the Komolgorov strong law of large numbers (Theorem 3.1).
4.
Proposition 2.11: Given
:,
kl
g R
R k l
and any sequence
n
b
such
that
b
b
s
a
n
.
.
where
n
b
and
b
are
1
k
vectors, if
g
is continuous at
b
, then
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View Full DocumentAEB 6933 – Mathematical Statistics for Food and Resource Economics
Lecture XV
Professor Charles B. Moss
Fall 2007
2
b
g
b
g
s
a
n
.
.
.
5.
Theorem 2.12: Suppose
a)
0
yX
;
b)
..
0
as
X
n
;
c)
XX
M
n
, finite and positive definite.
Then
n
exists
a.s.
for all
n
sufficiently large, and
0
n
.
d)
Proof: Since
X X n
M
, it follows from Proposition 2.11 that
det
det
X X n
M
. Because
M
is positive definite by (c),
det
0
M
. It follows that
det
0
X X n
a.s.
for all
n
sufficiently
large, so
1
X X n
exists
a.s.
for all
n
sufficiently large. Hence
n
y
X
n
X
X
n
'
'
ˆ
1
exists for all
n
sufficiently large.
In addition,
1
0
ˆ
'
'
n
X
nn
It follows from Proposition 2.11 that
0
1
0
.
.
0
0
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 Fall '09
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