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 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 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

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 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

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