Slides Chapter 4. Central limit and Slutsky's theorems

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Chapter 4
Central limit and Slusky's theorems
The central limit theorems (CLTs) give the asymptotic distribu-
tions of sums of independent random variables and Slutky's theo-
rems give the asymptotic distribution of functions of random vari-
ables and of sequences that are asymptotically equivalent to other
sequences.
4.1 Central limit theorems
Theorem 4.1 (Levy-Lindeberg's Theorem)
Let fX
n
g
n2IN
be a sequence of i.i.d.r.v.s with mean E(X
n
) = 
and variance V (X
n
) = 
2
,both nite.Then,
n
X
i=1
X
i
E

n
X
i=1
X
i
!
v
u
u
t
V

n
X
i=1
X
i
!
d
!N(0;1):
The rst known CLT was the theorembelow,due to De Moivre.
1
4.1.CENTRAL LIMIT THEOREMS
Corolary 4.1 (De Moivre's Theorem)
If fX
n
g
n2IN
is a sequence of i.i.d.r.v.s distributed as a Bern(p),
then
n
X
i=1
X
i
E

n
X
i=1
X
i
!
v
u
u
t
V

n
X
i=1
X
i
!
d
!N(0;1):
Example 4.1 (Normal approximation of binomial dis-
tribution)
Consider X
d
= Bin(n;p).We know that X =
P
n
i=1
Z
i
,where Z
i
are i.i.d.with Bern(p) distribution.By the De Moivre's Theorem,
n
X
i=1
Z
i
E

n
X
i=1
Z
i
!
v
u
u
t
V

n
X
i=1
Z
i
!
=
X np
p
np(1 p)
d
!N(0;1):
However,the normal approximation is poor whenever either n <
30 or n > 30 but p is too small (such that np < 5).
Theorem 4.2 (Lindeberg-Feller's CLT)
Let fX
n
g
n2IN
be a sequence of independent r.v.s with means E(X
n
) =

n
and variances V (X
n
) = 
2
n
,both nite 8n 2 N.Let c
2
n
=
P
n
i=1

2
i
.If the following condition,known as the Lindeberg-
Feller's condition (LFC),holds:
8 > 0;
1
c
2
n
n
X
i=1
E

(X
i

i
)
2
j jX
i

i
j   c
n

!
n!1
0;
ISABEL MOLINA 2
4.1.CENTRAL LIMIT THEOREMS
then
n
X
i=1
X
i
E

n
X
i=1
X
i
!
v
u
u
t
V

n
X
i=1
X
i
!
d
!N(0;1):
The previous theorem was also extended to sequences of trian-
gular arrays of r.v.s of the form:
X
11
X
21
X
22
X
31
X
32
X
33
  
;
where the r.v.s in each row are independent and satisfy the LFC,
see the theorem below.
Theorem 4.3 (Lindeberg-Feller's CLTfor triangular ar-
rays)
Let fX
ni
;i = 1;:::;ng
n2IN
be a sequence of triangular arrays of
r.v.s,where for each n 2 IN,the r.v.s in n-th row fX
n1
;:::;X
nn
g
are independent with nite means E(X
ni
) = 
ni
and variances
V (X
ni
) = 
2
ni
.Let c
2
n
=
P
n
i=1

2
ni
.If the LFC holds for each row,
that is,if
8 > 0;
1
c
2
n
n
X
i=1
E

(X
ni

ni
)
2
j jX
ni

ni
j   c
n

!
n!1
0;
ISABEL MOLINA 3
4.1.CENTRAL LIMIT THEOREMS
then
n
X
i=1
X
ni
E

n
X
i=1
X
ni
!
v
u
u
t
V

n
X
i=1
X
ni
!
d
!N(0;1):
Example 4.2 (Asymptotic normality of the LS estima-
tor is linear regression)
Consider the sequence of r.v.s dened as
X
i
=  +Z
i
+e
i
;i = 1;2;:::;
where Z
1
;Z
2
;:::are known xed values and e
1
;e
2
;:::are i.i.d.
r.v.s with E(e
i
) = 0 and V (e
i
) = 
2
,i = 1;2;:::.Let us dene
z
n
= n
1
P
n
i=1
Z
i
and s
2
n
= n
1
P
n
i=1
(Z
i
z
n
)
2
.The LS estimator
of  is given by
^

n
=
P
n
i=1
X
i
(Z
i
 z
n
)
P
n
i=1
(Z
i
 z
n
)
2
(a) See that
^

n
can be also expressed as
^

n
=  +
P
n
i=1
e
i
(Z
i
 z
n
)
P
n
i=1
(Z
i
 z
n
)
2
(b) Using (a) and applying the Lindeberg-Feller's CLT to the se-
quence of random variables
X
ni
= e
i
(Z
i
 z
n
);i = 1;:::;n;
prove that if

n
:= max
1in
(Z
i
 z
n
)
2
P
n
j=1
(Z
j
 z
n
)
2
!
n!1
0;
then
p
ns
n
(
^

n
)
d
!N(0;
2
):
ISABEL MOLINA 4
4.2.SLUTSKY'S THEOREMS
4.2 Slutsky's theorems
Theorem 4.4 (Slutsky's theorems)
Let fX
n
g
n2IN
be a sequence of d-dimensional r.v.s with X
n
d
!X.
Then it holds
(i) For any f:IR
d
!IR
k
such that P(X 2 C(f)) = 1,then
f(X
n
)
d
!f(X):
(ii) Let fY
n
g
n2IN
be another sequence of d-dimensional r.v.s with
X
n
Y
n
P
!0.Then,
Y
n
d
!X:
(iii) Let fY
n
g
n2IN
be another sequence of d-dimensional r.v.s with
Y
n
P
!c 2 IR
d
.Then,

X
n
Y
n

d
!

X
c

:
Example 4.3 Consider the sequence of r.v.s X
n
d
!N(0;1) and
the function f(x) = x
2
.Since f in continuous,by Theorem 4.4
(i),
X
2
n
d
!X
2
(1)
:
Example 4.4 Consider the sequence of r.v.s X
n
d
!N(0;1) and
the function f(x) = 1=x.Now f is not continuous at x = 0,but
since X is an absolutely continuous r.v.,P(X 2 C(f)) = P(X 2
f0g
c
) = 1 P(X = 0) = 1.Then,by Theorem 4.4 (i),
1=X
n
d
!1=X:
ISABEL MOLINA 5
4.2.SLUTSKY'S THEOREMS
Example 4.5 Consider the sequence of r.v.s X
n
= 1=n and the
function f = 1
(0;1)
.It holds that X
n
d
!0 but f(X
n
) = 1
d
9
f(X) = 0.This happens because C(f) = f0g
c
and P(X = 0) = 1.
Then,P(X 2 C(f)) = P(X 2 f0g
c
) = 0.
Corolary 4.2 (Asymptotic distribution of functions of
several sequences of random variables)
Let fX
n
g
n2IN
be a sequence of d-dimensional r.v.s with X
n
d
!X
and fY
n
g
n2IN
be a sequence of k-dimensional r.v.s with Y
n
P
!c 2
IR
k
.Let f:IR
d+k
!IR
r
be such that
P

X
c

2 C(f)

= 1:
Then it holds
f(X
n
;Y
n
)
d
!f(X;c):
ISABEL MOLINA 6