econ483lec01asymp - 1 Asymptotic Theory independent and...

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 ± , we have
1
p
n
n
P
i =1
X
i
° °
±
!