On a central limit theorem for a stationary multivariate linear process generated by linearly positive quadrant dependent random vectors

ON A CENTRAL LIMIT THEOREM FOR A STATIONARY MULTIVARIATE LINEAR PROCESS GENERATED BY LINEARLY POSITIVE

QUADRANT DEPENDENT RANDOM VECTORS

Tae-Sung Kim

Abstract. For a stationary multivariate linear process of the form X

_t

=

∞

X

j=0

A

j

Z

_t−j

, where {Z

t

: t = 0, ±1, ±2, · · · } is a sequence of stationary linearly positive quadrant dependent m-dimensional random vectors with E(Z

t

) = O and EkZ

t

k

²

< ∞, we prove a central limit theorem.

1. Introduction

Lehmman [8] introduced a simple and natural definition of positive dependence : A sequence {Y t : t = 0, 1, 2, · · · } 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 {Y i > α i , Y j > α j } ≥ P {Y i > α i }P {Y j >

α j }. A concept stronger than PQD was introduced by Newman [10]: A sequence {Y _t } of random variables is said to be linearly positive quadrant dependent (LPQD) if for any disjoint A, B and positive r _j ^′ s, X

i∈A

r i Y i and X

j∈B

r j Y j are PQD.

Two m-variate random vectors Z 1 , Z 2 are said to be positive quadrant dependent (PQD) if Z 1i , Z 2j are PQD for all i, j = 1, · · · , m, where Z 1i , Z 2 j are components of Z 1 , Z 2 , respectively.

Received August 28, 2000. Revised January 17, 2001.

2000 Mathematics Subject Classification: 60F05, 60G10.

Key words and phrases: multivariate linear process, linearly positive quadrant dependent random vectors, central limit theorem.

This work was supported by grant No. 2000-2-10400-001-3 from the Basic Re-

search Program of the Korea Science & Engineering Foundation.

Let (Z 1 , Z 2 , · · · , Z t ) be m-variate random vectors. We say that (Z 1 , Z ₂ , · · · , Z _t ) is linearly positive quadrant dependent if for any disjoint A, B ⊂ {1, · · · , t} and for any real vectors a r with nonnegative components,

X

s∈A

a s Z _s and X

r∈B

a r Z _r are PQD.

(1)

Let {X t , t = 0, ±1, · · · } be an m-variate linear process of the form X _t =

∞

X

u=0

A _u Z _t−u (2)

defined on a probability space (Ω, F, P ), where {Z _t } is a sequence of stationary m-variate LPQD random vectors with EZ t = O, EkZ t k ² < ∞ and positive definite covariance matrix Γ m×m . Throughout this paper we shall assume that

∞

X

u=0

kA _u k < ∞ and

∞

X

u=0

A _u 6= O _m×m , (3)

where for any m × m, m ≥ 1, matrix A = (a ij ), kAk =

m

X

i=1 m

X

j=1

|a ij | and O _m×m denotes the m × m zero matrix. Further, let

T =





∞

X

j=0

A j



 Γ





∞

X

j=0

A j





′

,

where the prime denotes transpose, and the matrix Γ = [σ kj ] with

σ _kj = E(Z 1 k Z 1j ) +

∞

X

t=2

(E(Z 1 k Z tj ) + E(Z 1j Z _tk )) . (4)

Further, let S n =

n

X

t=1

X _t , (n ≥ 0; S 0 = O).

Fakhre-Zakeri and Lee [4] proved a central limit theorem for multi- variate linear processes generated by independent multivariate random vectors and Fakhre-Zakeri and Lee [5] also derived a functional central limit theorem for multivariate linear processes generated by multivariate random vectors with martingale difference sequence.

In this note we prove a central limit theorem for an m-variate linear

process generated by m-variate LPQD random vectors.

Theorem 1.1. Let {Z t , t = 0, ±1, · · · } be a strictly stationary LPQD sequence of m-dimensional random vectors with E(Z _t ) = O, EkZ _t k ² <

∞ and positive definite covariance matrix Γ as in (4). Let {X t } be an m-variate linear process defined as in (2). Assume that

EkZ 1 k ² + 2

∞

X

t=2 m

X

i=1

E(Z 1i Z ti ) = σ ² < ∞, (5)

∞

X

t=n+1

EkZ 1 i Z _ti k = O(n ^−ρ ) for some ρ > 0, (6)

and

EkZ t k ^s < ∞ for some s > 2.

(7)

Then, the multivariate linear process {X t } fulfills the central limit the- orem, that is, n ⁻

¹²

S _n −→ N (O, T ). ^D

Remark. For m = 1, Kim and Baek [7] showed that the central limit theorem holds for the linear processes generated by an LPQD process.

2. Proofs

Note that Newman [10] has proved the central limit theorem for LPQD random variables (see Theorem 12 of [10]). Thus by means of the simple device due to Cramer Wold the following result holds.

Lemma 2.1. Let {Z t } be a sequence of stationary LPQD m-variate random vectors with E(Z _t ) = O and EkZ _t k ² < ∞. If (5) holds then

n ⁻

¹²

n

X

t=1

Z _t −→ N (O, Γ), ^D

where Γ = [σ _kj ] is defined as in (4); that is, {Z t } satisfies the central limit theorem.

Lemma 2.2. Let {Z _t } be a sequence of stationary LPQD random vectors with E(Z t ) = O, EkZ t k ² < ∞. Let ˜ X _t = (

∞

X

j=0

A j )Z t and ˜ S _k =

k

X

t=1

X ˜ _k . Assume that (5), (6) and (7) hold. Then

n ⁻

²¹

max

1≤k≤n k˜ S _k − S _k k = ◦ p (1).

(8)

Proof. See Appendix.

Proof of Theorem 1.1. As in Lemma 2.2, set ˜ X _t = (

∞

X

j=0

A j )Z t and S ˜ _n =

n

X

t=1

X ˜ _t . First note that

Ek ˜ X ₁ k ² + 2

∞

X

t=2 m

X

i=1

E( ˜ X 1i X ˜ ti )

= (

∞

X

j=1

A j ) ² (EkZ 1 k ² + 2

∞

X

t=2 m

X

i=1

E(Z 1 i Z ti )).

(9)

Since ˜ X _t is LPQD, by Lemma 2.1 { ˜ X _t } satisfies the central limit theorem, that is,

n ⁻

¹²

˜ S _n −→ N (O, T ). ^D (10)

According to Lemma 2.2 we also have n ⁻

¹²

|˜ S _n − S n | = ◦ p (1).

(11)

Hence from (10) and (11) the desired conclusion follows by Theorem 4.1 of [1].

Appendix

Proof of Lemma 2.2. We prove Lemma 2.2 using the ideas in the proof of Lemma 3 of [5] and Lemma 2 of [7]. First observe that

(A.1)

∞

X

t=n+1

E( ˜ X 1 i X ˜ _ti ) = (

∞

X

j=0

A _j ) ²

∞

X

t=n+1 m

X

i=1

E(Z 1 i Z _ti ) = O(n ^−ρ ) and that

(A.2) Ek ˜ X _t k ^s = (

∞

X

j=0

A j ) ^s EkZ t k ^s < ∞ for some s > 2.

By Lemma 3 of [7], it follows from (A.1) and (A.2) that

(A.3) E( max

1≥k<n k ˜ X ₁ + · · · + ˜ X _k k ^r ) ≥ Bn

^r2

for some r > 2 and a constant B.

Next, we observe that S ˜ _k =

k

X

t=1





k−t

X

j=0

A j



 Z _t +

k

X

t=1





∞

X

j=k−t+1

A j



 Z _t

=

k

X

t=1





t−1

X

j=0

A j Z _t−j



 +

k

X

t=1





∞

X

j=k−t+1

A j



 Z _t and thus

˜

S _k − S k = −

k

X

t=1

∞

X

j=t

A j Z _t−j +

k

X

t=1





∞

X

j=k−t+1

A j



 Z _t

= I 1 + I 2 (say).

To prove

(A.4) n ⁻

¹²

max

1≤k≤n kI 1 k −→ 0, ^P we observe that for r > 2

n ⁻

^2r

E max

1≤k≤n

k

X

t=1

∞

X

j=t

A j Z _t−j

r

= n ⁻

^2r

E max

1≤k≤n

∞

X

j=1 j∧k

X

t=1

A j Z _t−j

r

≤ n ⁻

^2r







∞

X

j=1

kA _j k





 E max

1≤k≤n

j∧k

X

t=1

Z _t−j

r 





1 r







r

≤ K





∞

X

j=1

kA j k  j ∧ k n



¹₂





r

for some positive constant K, where we have used Lemma 2 in [7] for

LPQD random variables. By the dominated convergence theorem the

last term above tends to zero as n −→ ∞ from which (A4) follows.

Next, we show that

(A.5) n ⁻

¹²

max

1≤k≤n kI 2 k = ◦ _p (1).

Write

I 2 = I 21 + I 22 , where

I 21 = A 1 Z _k + A 2 (Z _k + Z _k−1 ) + · · · + A _k (Z _k + · · · + Z 1 ) and

I 22 = (A k+1 + A k+2 + · · · ) (Z k + · · · + Z 1 ).

Let p n be a sequence of positive integers such that

(A.6) p n −→ ∞ and p n /n ^−→ 0 as n ^−→ ∞.

Note that n ⁻

¹²

max

1≤k≤n kI 22 k ≤

∞

X

i=0

kA i k

!

n ⁻

¹²

max

1≤k≤p

n

kZ 1 + · · · + Z k k

+



 X

i>p

n

kA i k



 n ⁻

¹²

max

1≤k≤n kZ 1 + · · · + Z k k

= III + IV (say).

It follows from (3) and (A.6) that for some r > 2 III ≤

∞

X

i=0

kA i k

! r

B 1 (p n /n)

^r2

−→ 0 ^P and

IV ≤



 X

i>p

n

kA i k





r

B 2

−→ 0, P

by Lemma 2 of [7]. It remains to prove that Y n : = n ⁻

¹²

max

1≤k≤n kI 21 k = ◦ p (1).

To this end, define for each l ≥ 1

I 21,l = B 1 Z _k + B 2 (Z k + Z k+1 ) + · · · + B k (Z k + · · · + Z 1 ), where

B _k =  A _k , k ≤ l

O _m×m , k > l.

Let Y _n,l = n ⁻

¹²

max _1≤k≤n kI 21 ,l k. Clearly, for each l ≥ 1,

(A.7) Y _n,l = ◦ _p (1).

On the other hand,

(Y _n,l − Y _n ) ≤ n ⁻

¹²

max

1≤k≤n

k

X

i=1

(A _i − B _i ) (Z _k + · · · + Z _k−i+1 )

≤ n ⁻

¹²

max

l<k≤n k

X

i=l+1

kA _i k max

l<i≤n kZ _k + · · · + Z _k−i+1 k

!

≤ n ⁻

¹²

X

i>l

kA i k max

l<k≤n max

l<i≤k (kZ 1 + · · · + Z _k k + kZ 1 + · · · + Z _k−i k)

≤ n ⁻

¹²

X

i>l

kA i k ( max

l<k≤n kZ 1 + · · · + Z _k | + max

l<k≤n max

l<i≤n kZ 1 + · · · + Z _k−i |)

≤ n ⁻

¹²

X

i>l

kA i k



1≤j≤n max kZ 1 + · · · + Z j k + max

1≤k≤n kZ 1 + · · · + Z j k



= 2n ⁻

¹²

X

i>l

kA _i k max

1≤j≤n kZ 1 + · · · + Z _j k.

From this result and Lemma 2 of [7], for any δ > 0,

(A.8)

l→∞ lim lim

n→∞ sup P (|Y n,l − Y n | ² > δ)

≤ lim

l→∞ 2 ^r δ ^−r X

i>l

kA i k

! r

n→∞ lim n ⁻

^r2

max

1≤j≤n kZ 1 + · · · + Z j k ^r

≤B lim

l→∞ δ ^−r 2 ^r X

i>l

kA i k

! r

= 0.

In view of (A.7) and (A.8), it follows from Theorem 4.2 of [1, p.25] that Y n = ◦ p (1). This completes the proof of Lemma 2.2.

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Division of Mathematical Science Wonkwang University

Ik-San, Chonbuk 570-749, Korea

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