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 (LevyLindeberg'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 (LindebergFeller'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 (LindebergFeller'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 nth 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 dened 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 dene
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 LindebergFeller'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
1in
(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 ddimensional 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 ddimensional 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 ddimensional 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 ddimensional r.v.s with X
n
d
!X
and fY
n
g
n2IN
be a sequence of kdimensional 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
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