Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2007, Article ID 74634, 7 pages
doi:10.1155/2007/74634
Research Article
Convergence of Weighted Linear Process for ρ-Mixing
Random Variables
Guang-Hui Cai
Received 10 April 2006; Revised 11 January 2007; Accepted 15 March 2007
A central limit theorem and a functional central limit
theorem are obtained for weighted
{Yi , 0 ≤ i < ∞} is a
linear process of ρ-mixing sequences for the Xt = ∞
i=0 ai Yt −i , where
i
sequence of ρ-mixing random variables with EYi = 0, 0 < EYi2 < ∞, ∞
i=1 ρ(2 ) < ∞. The
results obtained generalize the results of Liang et al. (2004) to ρ-mixing sequences.
Copyright © 2007 Guang-Hui Cai. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction and main results
A sequence {Xn , n ≥ 1} is said to be a ρ-mixing sequence, if n → ∞, we have
ρ(n) =
sup
∞
k≥1,X ∈L2 (F1k ),Y ∈L2 (Fk+n
)
cov(X,Y )/ X 2 Y 2 −→ 0,
(1.1)
where Fnm is a σ-field that is generated by the random variables Xn ,Xn+1 ,...,Xm . Here
X p = (E|X | p )1/ p .
We assume that {Yi , 0 ≤ i < ∞} is a ρ-mixing sequence. Let Xt be a linear process generated by Yt , that is,
Xt =
∞
ai Yt−i ,
(1.2)
i=0
where
∞
ai < ∞ .
i=0
(1.3)
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Discrete Dynamics in Nature and Society
And let EYi = 0, 0 < EYi2 < ∞, that is,
σ 2 = lim E
n
n→∞
2
n
i=1 Yi
> 0.
(1.4)
√
2
Let Sn = ni=1 Xi ,τ 2 = σ 2 ( ∞
i=0 ai ) ,ξn (u) = (S[nu] )/(τ n).
n
∞
√
Let Xt = ( i=0 ai )Yt , Sn = t=1 Xt , ξn (u) = (S[nu] )/(τ n).
For the linear process, Ho and Hsing [1], Phillips and Solo [2] and Wang et al. [3]
got central limit theorems (functional central limit theorems) for linear process under
independent assumptions. Kim and Baek [4] got a central limit theorem (functional central limit theorem) for strongly stationary linear process under linear positive quadrantdependent assumptions.
As for NA random variable, Liang et al. [5] obtained the following result.
Theorem
1.1. Let {Yn , n ≥ 0} be a sequence of NA random variables with EYi = 0 and
|
j:|k− j |≥u cov(Yk ,Y j )| → 0 as n → ∞ uniformly for k ≥ 1. Assume that {bni , 1 ≤ i ≤ n, n ≥
n 2
1} is an array of real numbers satisfying max1≤i≤n |bni | = O(n−1/2 ), satisfying i=1 bni
=
∞
n
O(1), max1≤i≤n |bni | → 0, as n → ∞ and that Var( i=1 bni Xi ) → 1. i=1 ρ(2i ) < ∞, {Yn2 ,
n ≥ 0} is uniformly integrable, (1.4) holds, and let {Yn , n ≥ 0} be a linear process defined by
(1.2). Suppose that (1.3) holds, then
n
∞
ai bni Xi −→ N 0,
2 d
i=1
.
(1.5)
i =0
We are inspired by Wang et al. [3] and Salvadori [6]. Salvadori [3] have obtained Linear combinations of order statistics to estimate the quantiles of generalized Pareto and
extreme values distributions. In this paper, we obtain a central limit theorem (functional
central limit theorem) for linear process under ρ-mixing sequence assumptions. The results obtained generalize the results of Liang et al. [5] to ρ-mixing sequences. More precisely, we will prove the following theorem.
sequence of indentically distributed random
Theorem 1.2. Let {Yn , n ≥ 0} be a ρ-mixing
i ) < ∞, {Y 2 , n ≥ 0} is uniformly integrable,
ρ(2
variables with EYi = 0, 0 < EYi2 < ∞, ∞
i =1
n
(1.4) holds, and let {Yn , n ≥ 0} be a linear process defined by (1.2). Suppose that (1.3) holds,
let {bni , 1 ≤ i ≤ n, n ≥ 1} is an array of real numbers satisfying max1≤i≤n |bni | = O(n−1/2 ),
then
n
i=1 bni Xi
τ
where Wn (t) =
[nt]
i=1
d
−→ N(0,1),
Wn =⇒ W,
(1.6)
√
Yi /σ n and {W(t); t ≥ 0} is a standard Brownian motion.
The weak convergence of Theorem 1.2 is quite useful in characterizing the limit distribution of various statistics arising from the inference of econometric theory, when the
economic time series { yt } defined as
yt = αyt−1 + Xt ,
t = 1,2,...,
(1.7)
Guang-Hui Cai 3
where y0 is a constant with probability one. The least squares estimator of α is given by
n
t =1 y t y t −1
α
n = n
2 .
(1.8)
t =1 y t
To test α = 1 against α < 1, that is, the unit root test, a key step is to derive the limit
distribution of the DF (Dickey-Feller) test statistic
n
t =1 yt −1 yt − yt −1
n−2 nt=1 yt2−1
n(α
n − 1) =
.
(1.9)
As shown by Phillips [7] and Wang et al. [3] obtained the limit distribution of n(α
n − 1)
when {Xt } is a linear process generated by an i.i.d. sequence under some conditions. In
this paper, we obtained the limit distribution of n(α
n − 1) when {Xt } is a linear process
under ρ-mixing random variables assumptions.
2. Proof of main theorem
In order to proof Theorem 1.2, we need the following lemmas.
Lemma 2.1 (see [8]). Let {Yn , n ≥ 1} be a centered ρ-mixing sequence, E|Yi | p < ∞, for some
p ≥ 2, then there exists a positive constant C = C(p,ρ(·)), such that
n
p
[logn]
2 p/2
i
E max Ym ≤ C exp C
ρ(2 ) n max E Ym
1≤m≤n
1≤m≤n
m=1
i =0
[logn]
+ nexp C
ρ
2/ p
i
(2 )
i =0
p
max EYm 1≤m≤n
(2.1)
.
Lemma 2.2 ([9]). Let {Yn , n≥1} be a ρ-mixing sequence with EYi =0, EYi2 <∞,
∞, {Yn2 , n ≥ 0} is uniformly integrable, and (1.4)holds, then
∞
i=1 ρ(2
i )<
n
i=1 Yi d
√
−→ N(0,1),
σ n
where Wn (t) =
[nt]
i=1
Wn =⇒ W,
(2.2)
√
Yi /σ n and {W(t); t ≥ 0} is a standard Brownian motion.
3. Proof of Theorem 1.2
It is clear that
Sk =
k
Xt =
t =1
=
k t
−1
t =1
i =0
k
t =1
k −t ai Yt +
i =0
ai Yt−i +
k
t =1
k
t =1
∞
i=k−t+1
∞
i=k−t+1
ai Yt .
ai Yt
(3.1)
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Discrete Dynamics in Nature and Society
Then
Sk − Sk = −
k ∞
t =1
ai Yt−i +
k
t =1
i =t
∞
ai Yt =: A + B.
(3.2)
i=k−t+1
First we proof
n−1/2 max Sk − Sk −→ 0.
P
(3.3)
1≤k≤n
In order to proof (3.3), we need only to show
P
(3.4)
P
(3.5)
n−1/2 max |A| −→ 0,
1≤k≤n
n−1/2 max |B | −→ 0.
1≤k≤n
Using the Minkowsky inequality, Lemma 2.1 with p = 2 and the dominated convergence
theorem, then
k ∞
n−1 E max 1≤k≤n
t =1 i =t
2
∞ i∧k
ai Yt−i = n−1 E max 1≤k≤n
−1
≤n E
≤ n −1
≤n
−1
∞
i=1
∞
i =1
i=1 t =1
2
ai Yt−i i ∧k
2
ai max Yt−i 1≤k≤n t =1
i ∧k
2 1/2 2
ai E max Yt−i 1≤k≤n ∞
ai 2
t =1
C exp C
i =1
≤ Cn−1
[logn]
(3.6)
i
ρ(2 ) (i ∧ n)EY12
i =0
∞
ai (i ∧ n)1/2
2
= o(1).
i=1
By (3.6), we have (3.4).
Because
B=
=
k
∞
t =1
i=k−t+1
k
k
i =1
ai
t =k−i+1
ai Yt
Yt +
∞
i=k+1
ai
k
t =1
(3.7)
Yt =: B1 + B2 .
Guang-Hui Cai 5
Let { pn } be a positive integers { pn } such that pn → ∞ and pn /n → 0, we have
n
−1/2
max B2 ≤
1≤k≤n
k Yi 1≤k≤ pn i =0
i=1
k ∞
ai n−1/2 max Yi +
1≤k≤n ∞
−1/2
ai n
max
i= pn +1
(3.8)
i =1
=: B21 + B22 .
Using Lemma 2.1 with p = 2, we have
2
E(B21 ) =
≤
∞
ai i =0
∞
ai 2
k 2
n E max Yi 1≤k≤ pn −1
i =1
2
n−1 C exp C
i =0
∞
ai ≤C
2 i=0
[logn]
ρ(2i ) pn EY12
(3.9)
i =0
pn
= o(1).
n
Using Lemma 2.1 with p = 2, we have
∞
2
E(B22 ) =
2
|ai |
i= pn +1
∞
≤
≤C
2
|ai |
∞
i=1
[logn]
−1
n C exp C
i= pn +1
k 2
n E max Yi 1≤k≤n −1
i
ρ(2 ) nEY12
(3.10)
i =0
2
= o(1).
|ai |
i= pn +1
By (3.8), (3.9), and (3.10), when n → ∞, we have
n−1/2 max B2 −→ 0.
P
1≤k≤n
(3.11)
Next, when n → ∞, we want to proof
Ln = n−1/2 max B1 −→ 0.
P
(3.12)
Yt ,
(3.13)
1≤k≤n
For each m ≥ 1, let
B1,m =
k
i=1
bi
k
t =k−i+1
where bi = ai I (i ≤ m). Let
Ln,m = n−1/2 max B1,m ,
1≤k≤n
(3.14)
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Discrete Dynamics in Nature and Society
for each m ≥ 1, when n → ∞, then
P
Ln,m ≤ a1 + · · · + am n−1/2 Y1 + · · · + Ym −→ 0,
(3.15)
for all ε > 0, by Lemma 2.1, we have
2
P Ln − Ln,m > ε ≤ ε−2 Ln − Ln,m
2
k
≤ ε−2 n−1 E max a
−
b
Y
+
·
·
·
+
Y
i
i
k
k−i+1 m≤k≤n i =1
≤ε
2
k
k
k
−i
ai Yi − Yi
−2 −1
n E max
m≤k≤n
∞
≤ 4ε−2
≤ 4ε
≤C
i=m+1
∞
−2
ai ai i=m+1
∞
ai i=m+1
i =1
i =1
2
k 2
−1
n E max Yi 1≤k≤n 2
(3.16)
i =1
−1
n C exp C
[logn]
i
ρ(2 ) nEY12
i=0
2
−→ 0,
i=m+1
when m → ∞. By (3.16), we have
Ln − Ln,m −P→ 0.
(3.17)
Using (3.15) and (3.17), we have (3.12). By (3.11), (3.12), and (3.7), we have (3.5).
Therefore we have (3.3). By max1≤i≤n |bni | = O(n−1/2 ), and (3.3), then
k
max bni Xi −
k
P
bni Xi −→ 0.
(3.18)
Sn d
−→ N(0,1),
τ
ξn =⇒ W,
(3.19)
Sn d
−→ N(0,1),
τ
ξn =⇒ W.
(3.20)
1≤k≤n
i =1
i =1
By Lemma 2.2 and (1.4), we have
and by (3.3), we have
Now, we complete the proof of Theorem 1.2.
Remark 3.1. Theorem 1.2 generalizes Theorem A to ρ-mixing sequences.
Guang-Hui Cai 7
Acknowledgments
The author would like to thank an anonymous referee for his/her valuable comments.
This paper is supported by Society Key Research Foundation of Zhejiang Province University (Statistics of Zhejiang Gongshang University) and Key discipline of Zhejiang
Province (Key discipline of Statistics of Zhejiang Gongshang University).
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Guang-Hui Cai: Department of Mathematics and Statistics, Zhejiang Gongshang University,
Hangzhou 310035, China
Email address: cghzju@163.com