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1 Asymptotic Theory
Assumption. Let
f
X
i
g
be a sequence of
independent and identically distributed (
iid
) random variables
with
EX
i
=
and
V ar
(
X
i
) =
±
2
(
and
±
2
are unknown) for
i
= 1
; :::; n
.
Theorem 1. Law of Large Numbers (LLN)
Under the above Assumption,
1
n
n
P
i
=1
(
X
i
)
!
p
0
;
as
n
! 1
:
Theorem 2. Central Limit Theorem (CLT)
Under the above Assumption,
1
p
n
n
P
i
=1
(
X
i
)
!
d
N
&
0
; ±
2
±
;
as
n
! 1
:
1.1 Law of Large Numbers (LLN)
1
n
n
P
i
=1
(
X
i
)
!
p
0
1. the LHS is random and the RHS is nonrandom
2. If we add
to both sides, we have
1
n
n
P
i
=1
X
i
!
p
.
3.
!
p
is a convergence concept that links a random variable to a nonrandom constant number.
4. The LLN says that in the limit (
n
! 1
),
the sum of random variables
X
i
, or the sum of centered random variables
(
X
i
)
weighted by
n
1
will be a nonrandom number.
1.2 Central Limit Theorem (CLT)
1
p
n
n
P
i
=1
(
X
i
)
!
d
N
&
0
; ±
2
±
1. both the LHS and RHS are random. however,
the distribution of the LHS is arbitrary
the distribution of the RHS is Normal with mean
0
and variance
±
2
2. If we divide both sides with
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This note was uploaded on 04/04/2011 for the course ECON 401 taught by Professor Staff during the Spring '08 term at University of Washington.
 Spring '08
 Staff
 Macroeconomics

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