A central limit theorem for general weighted sums of LPQD random variables and its application

A CENTRAL LIMIT THEOREM FOR GENERAL WEIGHTED SUMS OF LPQD RANDOM

VARIABLES AND ITS APPLICATION

Mi-Hwa Ko, Hyun-Chull Kim, and Tae-Sung Kim

Abstract. In this paper we derive the central limit theorem for P

i=1

a

ξ

, where {a

, 1 ≤ i ≤ n} is a triangular array of non- negative numbers such that sup

P

i=1

a

< ∞, max

a

→ 0 as n → ∞ and ξ

s are a linearly positive quadrant dependent sequence. We also apply this result to consider a central limit the- orem for a partial sum of a generalized linear process of the form X

= P

j=−∞

a

ξ

.

1. Introduction and results

Lehmann [7] introduced a simple and natural definition of positive dependence: A sequence {ξ _i , 1 ≤ i ≤ n} of random variables is said to be pairwise positive quadrant dependent (pairwise PQD) if for any real α _i , α _j and i 6= j P (ξ _i > α _i , ξ _j > α _j ) ≥ P (ξ _i > α _i )P (ξ _j > α _j ).

Much stronger concepts than PQD was considered by Esary, Proschan and Walkup [4]: A sequence {ξ _i , 1 ≤ i ≤ n} of random variables is said to be associated if for any real coordinatewise increasing functions f, g on R ⁿ , Cov(f (ξ ₁ , . . . , ξ _n ), g(ξ ₁ , . . . , ξ _n )) ≥ 0.

Instead of association, Newman’s [9] central limit theorem requires only that positive linear combinations of the random variables are PQD.

The definition of positive dependence introduced by Newman [9] is the following: A sequence {ξ _i , 1 ≤ i ≤ n} of random variables is said to be linearly positive quadrant dependent(LPQD) if for every pair of disjoint

Received December 21, 2004.

2000 Mathematics Subject Classification: 60F05, 60G10.

Key words and phrases: central limit theorem, linear process, linearly positive quadrant dependent, uniformly integrable, triangular array.

This work was supported by WonKwang University in 2006.

subsets A, B ⊂ {1, 2, . . . , n} and positive r ⁰ _j s

(1.1) X

i∈A

r _i ξ _i and X

j∈B

r _j ξ _j are P QD.

Let us remark that LPQD is between pairwise PQD and association and it is well known(see, for example, Newman [9, p.131] that association implies LPQD and LPQD implies PQD.

Newman [9] established the central limit theorem for a strictly sta- tionary LPQD process and Birkel [2] also obtained a functional central limit theorem for LPQD processes. Kim and Baek [6] extended this re- sult to a stationary linear process of the form Y _k = P _∞

j=0 a _j ξ _k−j , where {a _j } is a sequence of real numbers with P _∞

j=0 |a _j | < ∞ and {ξ _k } is a strictly stationary LPQD process with Eξ _i = 0 and 0 < Eξ _i ² < ∞.

In this paper we derive a central limit theorem for a linearly positive quadrant dependent sequence in a double array, replacing the strictly stationarity assumption with uniform integrability (see Theorem 1.1 be- low). We apply this result to obtain a central limit theorem for a partial sum of a linear process X _n = P _∞

j=−∞ a _k+j ξ _j generated by linearly pos- itive quadrant dependent sequence {ξ _j } (see Theorem 1.2 below).

Newman [9] proved that the following central limit theorem for strictly stationary LPQD sequence:

Theorem A. [9] Let {ξ _i , i ≥ 1} be a strictly stationary sequence of LPQD random variables with Eξ _i = 0 and Eξ _i ² < ∞. If we assume that

σ ² = X ∞ i=1

Cov(ξ ₁ , ξ _i ) < ∞, then

(σ ² n) ⁻

X n

i=1

ξ _i → ^D N (0, 1) as n → ∞.

The next theorem extends Newman’s central limit theorem for a strictly stationary LPQD sequence (Theorem A) from equal weights to general weights, while at the same time weakening the assumption of stationarity.

Theorem 1.1. Let {a _ni , 1 ≤ i ≤ n} be a triangular array of non- negative numbers such that

(1.2) sup

n

X n i=1

a ² _ni < ∞,

and

(1.3) max

1≤i≤n a _ni → 0 as n → ∞.

Let {ξ _i } be a centered sequence of linearly positive quadrant depen- dent random variables such that

(1.4) {ξ _i ² } is an uniformly integrable family,

(1.5) Var(

X n i=1

a _ni ξ _i ) = 1 and

(1.6) X

j:|k−j|≥u

Cov(ξ _k , ξ _j ) → 0 as u → ∞ uniformly in k ≥ 1 (see Cox and Grimmet [3]). Then

X n i=1

a _ni ξ _i −→ N(0, 1) as n → ∞. ^D

Corollary 1.1. Let {ξ _i } be a centered sequence of linearly positive quadrant dependent random variables such that {ξ ² _i } is an uniformly integrable family and let {a _ni , 1 ≤ i ≤ n} be a triangular array of nonnegative numbers such that

(1.7) sup

n

X n i=1

a ² _ni σ _n ² < ∞,

(1.8) max

1≤i≤n

a _ni

σ _n → 0 as n → ∞, where σ _n ² = Var( P _n

i=1 a _ni ξ _i ). If (1.6) holds then, as n → ∞

(1.9) 1

σ _n X n

i=1

a _ni ξ _i −→ N(0, 1). ^D

Theorem 1.2. Let {a _j , j ∈ Z} be a sequence of nonnegative numbers such that P

j a _j < ∞ and let {ξ _j , j ∈ Z} be a centered sequence of lin- early positive quadrant dependent random variables which is uniformly integrable in L ₂ and satisfies (1.6). Let

X _k = X ∞ j=−∞

a _k+j ξ _j and S _n = X n i=1

X _i .

Assume

(1.10) inf

n≥1 n ⁻¹ σ _n ² > 0, where σ _n ² = Var(S _n ). Then

(1.11) S _n

σ _n

−→ N(0, 1) as n → ∞. D

This result extends Theorem 18.6.5 in Ibragimov and Linnik [5] from the i.i.d. case to the linearly positive quadrant dependence case by adding condition (1.6) and improves on the central limit theorem of Kim and Baek [6] for linear process generated by LPQD sequence.

2. Proofs

We start with the following lemma.

Lemma 2.1. [8] Let {Z _i , 1 ≤ i ≤ n} be a sequence of linearly positive quadrant dependent random variables with finite second moments. Then

¯ ¯

¯ ¯E exp(it X n j=1

Z _j ) − Y n j=1

E exp(itZ _j )

¯ ¯

¯ ¯ ≤ Ct ²

¯ ¯

¯ ¯ Var(

X n j=1

Z _j ) − X n j=1

Var(Z _j )

¯ ¯

¯ ¯

for all t ∈ R, where C > 0 is an arbitrary constant, not depending on n.

Proof of Theorem 1.1. Without loss of generality we assume that a _ni = 0 for all i > n. For every 1 ≤ a < b ≤ n and 1 ≤ u ≤ b − a we have, after simple manipulations,

(2.1)

0 ≤

b−u X

i=a

a _ni X b j=i+u

a _nj Cov(ξ _i , ξ _j )

≤ sup

k

µ X

j:|k−j|≥u

Cov(ξ _k , ξ _j )

¶µ X _b

i=a

a ² _ni

¶ .

In particular by (1.6), we have

(2.2) sup

k

µ X

j:|k−j|≥1

Cov(ξ _k , ξ _j )

¶

< ∞.

Without restricting the generality we can assume that sup _k Eξ ² _k = 1.

Hence, there exists a constant C > 0 such that for every 1 ≤ a ≤ b

≤ n,

(2.3) Var

µ X _b

i=a

a _ni ξ _i

¶

≤ X b i=a

a ² _ni Var(ξ _i ) + 2 X b−1 i=a

a _ni X b j=i+1

a _nj Cov(ξ _i , ξ _j )

≤ X b i=a

a ² _ni Var(ξ _i )

+ 2 sup

k

µ X _∞

j:|k−j|≥1

Cov(ξ _k , ξ _j )

¶µ X _b

i=a

a ² _ni

¶

≤ C µ X _b

i=a

a ² _ni

¶

by (2.1) and (2.2).

We shall construct now a triangular array of random variables {Z _ni , 1

≤ i ≤ n} for which we shall make use of Lemma 2.1. Fix a small positive

² and find a positive integer u = u _² such that, for every n ≥ u + 1

(2.4) 0 ≤

µ _n−u X

i=1

a _ni X n j=i+u

a _nj Cov(ξ _i , ξ _j )

¶

≤ ².

This is possible because of (2.1) and (1.6). Denote by [x] the integer part of x and define

K =

· 1

²

¸ ,

Y _nj =

u(j+1) X

i=uj+1

a _ni ξ _i , j = 0, 1, . . . ,

A _j =

½

i : 2Kj ≤ i < 2Kj + K, Cov(Y _ni , Y _n,i+1 )

≤ 2 K

2Kj+K X

i=2Kj

Var(Y _ni )

¾ .

Since 2Cov(Y _ni , Y _n,i+1 ) ≤ Var(Y _ni )+Var(Y _n,i+1 ), we get that for every j the set A _j is not empty. Now we define the integers m ₁ , m ₂ , . . . , m _n recursively. Let m ₀ = 0 and

m _j+1 = min{m : m > m _j , m ∈ A _j }

and define

Z _nj =

m X

i=m

+1

Y _ni , j = 0, 1, . . . ,

∆ _j = {u(m _j + 1) + 1, . . . , u(m _j+1 + 1)}.

We observe that

Z _nj = X

k∈∆

a _nk ξ _k , j = 0, 1, . . . .

By the definition of LPQD the sequence {Z _nj } is LPQD. From the fact that m _j ≥ 2K(j −1) and m _j+1 ≤ K(2j +1) every set ∆ _j contains no more than 3 Ku elements and m _j+1 /m _j → 1 as j → ∞. Hence, for every fixed positive ² by (1.2)–(1.5) the array {Z _ni : i = 0, 1, . . . , n; n ≥ 1}

satisfies the Lindeberg condition(see Petrov [10], Theorem 22, p.100), that is, {Z _nj } satisfies

(2.5) σ ⁻¹ _n X n j=1

EZ _nj ² I(|Z _nj | > ²σ _n ) → 0 as n → ∞,

where σ _n ² = Var( P _n

j=1 Z _nj ). We can observe that by Lemma 2.1 and the construction,

(2.6) ¯

¯ ¯

¯E exp(it X n j=1

Z _nj ) − Y n j=1

E exp(itZ _nj )

¯ ¯

¯ ¯

≤ Ct ²

¯ ¯

¯ ¯

½ Var(

X n j=1

Z _nj ) − X n j=1

Var(Z _nj )

¯ ¯

¯ ¯

¾

≤ Ct ²

½ 2

µ X _n

i=1

Cov(Z _ni , Z _n,i+1 )

¶ + 2

µ _n−2 X

i=1

X n j=i+2

Cov(Z _ni , Z _nj )

¶¾

≤ Ct ²

·½ 2

X n j=1

Cov(Y _n,m

, Y _n,m

₊₁ ) + 2

n−u X

i=1

a _ni X n j=i+u

a _nj Cov(ξ _i , ξ _j )

¾

+

½ 2

n−u X

i=1

a _ni X n j=i+u

a _nj Cov(ξ _i , ξ _j )

¾¸

= Ct ²

½ 4

n−u X

i=1

a _ni X n j=i+u

a _nj Cov(ξ _i , ξ _j ) + 2 X n j=1

Cov(Y _n,m

, Y _n,m

₊₁ )

¾

≤ Ct ²

½ 4² + 8

K X n i=1

Var(Y _ni )

¾

= Ct ²

½ 4² + 8

K X n j=1

Var(

u(j+1) X

i=uj+1

a _n

ξ _i )

¾

≤ Ct ²

½

4² + 8M K

X n j=1

u(j+1) X

i=uj+1

a ² _n

¾ ¡

by (2.3) ¢

≤ C ₁ t ² ²

½

1 + sup

n

X n i=1

a ² _ni

¾

≤ C ₂ t ² ² for every positive ².

Therefore the problem is now reduced to the study of the central limit theorem of a decoupled sequence { ˜ Z _nj } of independent random variables such that for each n and j, the variable ˜ Z _nj is distributed as Z _nj .

By (2.5), { ˜ Z _nj } also satisfies the Lindeberg condition, that is, { ˜ Z _nj } satisfies ˜ σ _n ⁻¹ P _n

j=1 E ˜ Z _nj ² I(| ˜ Z _nj | > ²˜ σ _n ) → 0 as n → ∞ , where ˜ σ _n ² = Var( P _n

j=1 Z ˜ _nj ), and hence by Theorem 7.2 of Billingsley [1]

(2.7) σ ˜ _n ⁻¹ X n j=1

Z ˜ _nj −→ N (0, 1) as n → ∞, ^D

where ˜ σ _n ² = Var( P _n

j=1 Z ˜ _nj ). It follows from (2.5), (2.6), and (2.7) that

(2.8) σ ⁻¹ _n

X n j=1

Z _nj −→ N (0, 1) as n → ∞, ^D where σ ² _n = Var( P _n

j=1 Z _nj ), and now the proof is complete by (2.7), (2.8), and Theorem 4.2 in Billingsley [1].

Proof of Corollary 1.1. Let A _ni = ^a _σ

n

. Then we have

1≤i≤n max A _ni → 0 as n → ∞, sup

n

X n i=1

A ² _ni < ∞,

Var(

X n i=1

A _ni ξ _i ) = 1.

Hence, by Theorem 1.1 the desired result (1.11) follows.

Proof of Theorem 1.2. First note that P

j a ² _j < ∞ and, without loss of generality, we can assume that sup Eξ _k ² = 1. Let

S _n = X n k=1

X _k = X ∞ j=−∞

µ X _n

k=1

a _k+j

¶ ξ _j . In order to apply Theorem 1.1, fix W _n such that P

|j|>W

a ² _j < n ⁻³ and take k _n = W _n + n. Then

S _n

σ _n = X

|j|≤k

µ X _n

k=1

a _k+j

¶ ξ _j

σ _n + X

|j|>k

µ X _n

k=1

a _k+j

¶ ξ _j

σ _n = T _n + U _n . By the Cauchy Schwarz inequality and the assumptions we have the following estimate

Var(U _n ) ≤ C X

|j|>k

Var µ X _n

k=1

a _k+j ξ _j σ _n

¶

≤ C X

|j|>k

µ X _n

k=1

a _k+j /σ _n

¶ ₂ Eξ ² _j

≤ Cnσ _n ⁻² X

|j|>k

µ X _n

k=1

a ² _k+j

¶

≤ Cn ² σ _n ⁻² X

|j|>k

−n

a ² _j

≤ Cn ² σ _n ⁻² X

|j|>W

a ² _j

≤ Cn ⁻¹ σ ⁻² _n → 0 as n → ∞ which yields

(2.9) U _n → 0 in probability as n → ∞.

By Theorem 4.1 of Billingsley [1], it remains to prove that T _n −→ ^D N(0, 1). Put

(2.10) a _nk =

P _n

j=1 a _k+j σ _n . From the assumption P

j a _j < ∞ (a _j > 0), (1.10), and (2.10) we obtain sup −∞<k<∞

P _n

j=1 a _k+j

σ _n → 0 as n → ∞,

1≤k≤n max a _nk → 0 as n → ∞,

sup

n

X n k=1

a ² _nk < ∞. (see Remark below.) Hence, by Theorem 1.1

(2.11) T _n −→ N(0, 1) ^D

and from (2.9) and (2.11) the desired result (1.10) follows.

Remark. In the proof of Theorem 1.2, let us suppose on the contrary that for some ² > 0 there exists a subsequence (j

, n

), n

→ ∞ such that P _n

k=1 a _k+j

> ²σ _n

. Denote by A = sup −∞<k<∞ a _k and notice that for r > j

n

X

k=1

a _k+r > ²σ

_n − 2A(r − j).

Hence σ ² _n

b ≥

j X

+w i=j

µ X _n

k=1

a _k+j

¶ ₂

≥ W ² ² σ _n ²

− 4Aσ _n

² µ j X

+W

i=j

(i − j

)

¶

≥ W ² ² σ _n ²

− 4Aσ _n

²W ² .

Taking W to be the least integer greater than or equal to _b² ³

and because σ _n

→ ∞, we obtain for n

sufficiently large,

σ _n ²

b ≥ 3σ _n ²

b − σ _n

36A

b ² ² ³ ≥ 2σ _n ²

b which is a contradiction.

Acknowledgements. The authors are exceptionally grateful to the referee for offering helpful remarks and comments that improved presentation of the paper.

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Mi-Hwa Ko

Statistical Research Center for Complex Systems Seoul National University

Seoul 151-742, Korea

E-mail: [email protected] Hyun-Chull Kim

Department of Mathematics Education Daebul University

Chonnam 526-702, Korea

E-mail: [email protected] Tae-Sung Kim

Department of Mathematics and Institute of Basic Science WonKwang University

Jeonju 570-749, Korea

E-mail: [email protected]