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Central limit theorem for stationary linear processes generated by linearly negative quadrant-dependent sequence
Journal of Inequalities and Applications volume 2012, Article number: 45 (2012)
Abstract
A central limit theorem is obtained for stationary linear process of the form , where {ε i } is strictly stationary sequence of linearly negative quadrant dependent random variables with Eε i = 0, for some s > 2, and .
1 Introduction and main results
Lehmann [1] introduced a new definition of negative dependence: negative quadrant dependent sequence (NQD). A concept stronger than NQD was introduced by Newman [2] that is said to be linear negative quadrant dependent (LNQD).
Definition 1.1. (Cf. Lehmann [1]). Two random variables X and Y are said to be negative quadrant dependent (NQD, in short) if for any x, y ∈ ℝ,
A sequence {X n }n∈ℕof random variables is said to be pairwise NQD, if X i and X j are NQD for all i, j ∈ ℕ+ and i ≠ j.
Definition 1.2. (Cf. Newman [2]). A sequence {X n }n∈ℕof random variables is said to be linearly negative quadrant dependent (LNQD, in short) if for any disjoint subsets A, B ∈ ℤ+ and positive ,
Remark 1.3. It is easily seen that if {X n }n∈ℕis a sequence of LNQD random variables, then {aX n + b}n∈ℕis still a sequence of LNQD random variables, where a and b are real numbers.
Example 1. Consider three discrete random variables with joint density, P(x, y, z) = P(X1 = x, X2 = y, X3 = z) with P(1, 3, 1) = P(2, 3, 1) = P(3, 2, 1) = P(3, 3, 1) = 1/11, P(3, 3, 2) = 7/11. We could show that (X1, X2, X3) is LNQD sequence.
Note that some applications for LNQD sequence have been found. See for example, Newman [2] established the central limit theorem for a strictly stationary LNQD process. Wang and Zhang [3] provided uniform rates of convergence in the central limit theorem for LNQD sequence. Ko and Choi [4] obtained the Hoeffding-type inequality for LNQD sequence. Fu and Wu [5] studied the almost sure central limit theorem for LNQD sequences, and so forth.
The aim of this article is to establish a central limit theorem for stationary linear process generated by LNQD random variables.
Let {X n }n∈ℕbe a stationary process of the form below defined on a probability space (Ω, , P)
where {a j } is a sequence of real numbers with and {ε i } is a strictly stationary sequence such that Eε i = 0, .
The linear processes are of special importance in time series analysis and they arise in wide variety of concepts (see, e.g., Hannan [[6], Chap. 6]). Applications to economics, engineering, and physical science are extremely broad and a vast amount of literature is devoted to the study of the theorems for linear processes under various conditions on ε t . For the linear processes, Fakhre-Zakeri and Lee [7] and Fakhre-Zakeri and Farshidi [8] established CLT under the i.i.d. assumption on ε t and Fakhre-Zakeri and Lee [9] proved a FCLT under the strong mixing condition on ε t . After Birkel [10] gave Lemma 3 which could be used to prove an inequality the same as the form of Doob's maximal inequality, Kim and Baek [11] obtained a central limit theorem for stationary linear processes generated by linearly positively quadrant dependent process. Peligrad and Utev [12] established the central limit theorem for linear processes with dependent innovations including martingales and mixingale.
Let . Define for n ≥ 1 the stochastic process
where [x] is the greatest integer not exceeding x.
Since LNQD is much weaker than NA, and NA is much weaker than independent case, studying the central limit theorems for LNQD fields is of interest. In this article, we establish a CLT (FCLT) for strictly stationary linear process of the form (1.4), generated by an LNQD sequence {ε i }. More precisely, we will prove the following theorems.
Theorem 1.4. Let {X t } be a stationary linear process of the form (1.3), where {a j } is a sequence of constants with and {ε i } is a strictly stationary LNQD sequence with Eε i = 0. Assume
Then the linear process {X t } fulfills the CLT.
Example 2. Let be a strictly stationary linear process generated by LNQD sequence, where ε t is a strictly stationary LNQD sequence random variables with Eε t = 0 and E|ε t |s < ∞ for some s > 2 satisfying (1.5) and (1.6). Here we suppose that the linear operators a j are geometrically bounded, in the sense that there exist real constants b > 0 and 0 < ρ < 1 such that a j ≤ bρjfor all j > 0. Then this process satisfies Theorem 1.4.
Theorem 1.5. Let {X t } be a stationary linear process of the form (1.3), defined in Theorem 14. If (1.5) and (1.6) are fulfilled, then the process {ξ n } satisfies the FCLT; that is, the process {ξ n } converges weakly to Wiener measure W on the space of all functions on [0,1], which have left-hand limits and are continuous from the right.
2 Proofs
The following lemma is needed to prove Theorems 1.4 and 1.5 and it is established by modifying the proof of Lemma 3 in Fakhre-Zakeri and Lee [9]. Doob's maximal inequality has played an important role in their proof. However, in our case, Doob's maximal inequality cannot be used, instead the inequality E(max1≤k≤n|ε1 + ⋯ + ε k |r) ≤ Bnr/2for r > 2 and B as in Lemma 2.2 will be used.
Lemma 2.1. Let {ε i } be a strictly stationary LNQD sequence with mean zero and . Let , and let , where {a j } is a sequence of real numbers with . If (1.5) and (1.6) are fulfilled, then
We close this section by introducing a maximal inequality which is needed to prove Lemma 2.1.
Lemma 2.2. Let {ε t } be a strictly stationary LNQD sequence with mean zero and . Assume that (1.6) and (1.7) hold. Then there exist r > 2 and a constant B not depending on n such that for n ≥ 1
According to Lemma 3.4 of Zhang [13], we can easily obtain the result.
Proof of Lemma 2.1. As in the proof of Lemma 3 of Fakhre-Zakeri and Lee [9], we have
Thus
It suffices to prove
and
First, by Minkowski's inequality and Lemma 2.2 we have for r > 2
Hence (2.5) is proved by the Markov inequality. To prove (2.6), write II = IIk 1+ IIk 2, where IIk 1= a1ε k + a2(ε k + εk-1) + ⋯ + a k (ε k + ⋯ + ε1) and IIk 2= (ak+1+ ak+2+ ⋯ )(ε k + ⋯ + ε1), and let {p n } be a sequence of positive integers such that
Then
It follows from (2.2), (2.8), that for r > 2 and a constant B1,
and thus by the Markov inequality. Similarly, by assumption for r > 2 and a constant B2,
and thus by the Markov inequality. Hence, . It remains to show that . For each m ≥ 1, define IIk 1,m= b1ε k + b2(ε k + εk-1) + ⋯ + b k (ε k + ⋯ + ε1), where b k = a k for k ≤ n and b k = 0 otherwise, and let Ln,m= n-1/2 max1≤k≤n|IIk 1,m|. Then
for each m, and
Since
the right-hand side of (2.13) is
Therefore, it follows from (2.2), (2.12), (2.13) and the Markov inequality that for any δ > 0,
In view of (2.16) and (2.15), it follows from Theorem 4.2 Billingsley [[14], p. 25].
Proof of Theorem 1.4. As in Lemma 2.1, set , and . Then by (1.4), we have
Since form a stationary LNQD sequence, satisfies the CLT by Theorem 12 of Newman [2]; that is,
According to Lemma 2.1, we also have
Hence from (2.18) and (2.19) the desired conclusion follows.
Proof of Theorem 1.5. Note that {} is a stationary LNQD process and that,
Thus it follows from Lemma 2.2 and (2.20), that {} satisfies the conditions of Corollary 2 of Birkel [10]. This implies that Theorem 1.5 holds for the sequence {}, where we define as in (1.4), with replacing S r . By Lemma 2.1, for all 0 ≤ u ≤ 1. Hence, the desired conclusion follows.
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Acknowledgements
This study was supported by the National Natural Science Foundation of China (11061012), the Guangxi China Science Foundation (2011GXNSFA018147) and Innovation Project of Guangxi Graduate Education (2010105960202M32). We are very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper.
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QW participated in the design of the study. JW conceived of the study, and participated in the design and proof of the article. All authors read and approved the final manuscript.
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Wang, J., Wu, Q. Central limit theorem for stationary linear processes generated by linearly negative quadrant-dependent sequence. J Inequal Appl 2012, 45 (2012). https://doi.org/10.1186/1029-242X-2012-45
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DOI: https://doi.org/10.1186/1029-242X-2012-45