The law of the iterated logarithm for LNQD sequences

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{\xi_{i},i\in{\mathbb{Z}}\}$\end{document}{ξi,i∈Z} be a stationary LNQD sequence of random variables with zero means and finite variance. In this paper, by the Kolmogorov type maximal inequality and Stein’s method, we establish the result of the law of the iterated logarithm for LNQD sequence with less restriction of moment conditions. We also prove the law of the iterated logarithm for a linear process generated by an LNQD sequence with the coefficients satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sum_{i=-\infty}^{\infty}|a_{i}|<\infty$\end{document}∑i=−∞∞|ai|<∞ by a Beveridge and Nelson decomposition.


Introduction
Two random variables X and Y are said to be negatively quadrant dependent (NQD, for short), if P(X ≤ x, Y ≤ y) -P(X ≤ x)P(Y ≤ y) ≤ 0 for all x, y ∈ R. A sequence {X k , k ∈ Z} is said to be linear negatively quadrant dependent (LNQD, for short) if for any disjoint finite subsets A, B ⊂ Z and any positive real numbers r j , i∈A r i X i and j∈B r j X j are NQD. It is obvious that LNQD implies NQD. The definitions of NQD and LNQD can be found in Lehmann [1] and Newman [2].
A much stronger concept than LNQD was introduced by Joag-Dev and Proschan [3]: for a finite index set I, the r.v.s. {X i , i ∈ I} are said to be negatively associated (NA, for short), if for any disjoint nonempty subsets A and B of I, and any coordinatewise nondecreasing function G and H with G : R A → R and H : R B → R and EG 2 (X i , i ∈ A) < ∞, EH 2 (X j , j ∈ B) < ∞, we have Cov(G(X i , i ∈ A), H(X j , j ∈ B)) ≤ 0. An infinite family is NA if every finite subfamily is NA.
Some applications for LNQD sequence have been found. For example, Newman [2] established the central limit theorem for a strictly stationary LNQD process, Dong and Yang [4] provided the almost sure central limit theorem for an LNQD sequence, Wang and Zhang [5] provided uniform rates of convergence in the central limit theorem for LNQD sequence, Li and Wang [6] obtained the asymptotic distribution for products sums of LNQD sequence, Ko et al. [7] studied the strong convergence for weighted sums of LNQD arrays, Ko et al. [8] obtained the Hoeffding-type inequality for LNQD sequence, Zhang et al. [9] established an almost sure central limit theorem for products sums of partial sums under LNQD sequence, Wang et al. [10] discussed the exponential inequalities and complete convergence for an LNQD sequence, Choi [11] obtained the Limsup results and a uniform LIL for partial sums of an LNQD sequence, Wang and Wu [12] obtained the strong laws of large numbers for arrays of rowwise NA and LNQD random variables, Wang and Wu [13] established the central limit theorem for stationary linear processes generated by LNQD sequence, Li et al. [14] established some inequalities for LNQD sequence, Shen et al. [15] proved the complete convergence for weighted sums of LNQD sequence, and so forth. It is easily seen that independent random variables and NA random variables are LNQD. Since LNQD random variables are much weaker than independent random variables and NA random variables, studying the limit theorems for LNQD sequence is of interest.
The main purpose of this paper is to discuss the limit theory for LNQD sequence. In Section 2, by the Kolmogorov type maximal inequalities and Stein's method, we obtain the law of the iterated logarithm for strictly stationary LNQD sequence with finite variance. In Section 3, we prove the law of the iterated logarithm for linear process generated by LNQD sequence with less restrictions by Beveridge and Nelson decomposition for linear process.
Throughout the paper, C denotes a positive constant, which may take different values whenever it appears in different expressions. We have log x = ln max{e, x}.

Main results
We will need the following property.
(H1) (Hoeffding equality): For any absolutely continuous functions f and g on R 1 and for any random variables X and Y satisfying Ef Now we state the law of iterated logarithm for LNQD sequence.
In particular, we have . (2.4) Proof By Lemma 2.3, following the proof of Theorem 3 in Shao [16], we can easily get the results of Lemma 2.4.

Lemma 2.5
Let {Y i , 1 ≤ i ≤ n} be an LNQD sequence of random variables with EY i = 0 and E|Y i | 3 < ∞. Define T n = n i=1 Y i and B 2 n = n i=1 EY 2 i . Then, for any x > 0, where is the standard normal distribution function.
Proof We will apply the Stein method. Let X be a standard normal random variable and define Let f be the unique bounded solution of the Stein equation The solution f is given by It is well known that (see Stein [17]) where By the definition of LNQD, we know ζ i and W (i) are NQD, then by (H1) and (2.6) we have By (2.6), we obtain For fixed 0 < t < 1, xI{0 ≤ t ≤ x} is a nondecreasing functions of x, by the definition of LNQD and Lemma 2.3, ζ i,1 I{0 ≤ t ≤ ζ i,1 } and W (i) are NQD. Then by (H1) and (2.7), Similarly, It follows from (2.6) that Finally, by putting the above inequalities together, we complete the proof of Lemma 2.5.
Proof of Theorem 2.1 It suffices to show that for 0 < ε < 1 Let m be an integer such that It is obvious that S n = S n,1 + S n,2 . By the same argument as of equation (2.2) from de Acosta [18], it is easy to check that Hence, by Kronecker's lemma and S n,2 /(n log log n) 1/2 → 0 a.s. (2.12) Observe that for every n sufficiently large, Hence, by (2.10) provided that n is sufficiently large. By the definition of LNQD and Lemma 2.3, we know {u i , i ≥ 1} are also LNQD random variables with Eu i = 0 and |u i | ≤ 2ma im for every i. By Lemma 2.4 (with α = 1ε, a = 2ma n ), (2.13) and (2.14), we get P max 1≤i≤n |S i,1 | ≥ (1 + 8ε) 2σ 2 n log log n To prove (2.9), let It suffices to show that ∞ k=1 P S n k ,1 ≥ (1 -7ε) 2σ 2 n k log log n k 1/2 = ∞. (2.17) In fact, by Lemma 2.4, similar to the proof of (2.15), we obtain ∞ k=1 P S n k-1 ,1 ≥ ε 2σ 2 n k log log n k 1/2 < ∞.
It is easy to see that From Lemma 2.5, we obtain P T k,1 ≥ (1 -6ε) 2σ 2 n k log log n k where Obviously, we have ∞ k=1 1 -1 + (1 -5ε)(2 log log n k ) 1/2 = ∞. (2.21) Noting that {v i,1 , 1 ≤ i ≤ k 4 } is an LNQD sequence and by (H1), we get By the fact that n k-1 = o(p k ), we see that Finally, we estimate J k,2 . By the Rosenthal type maximal inequality for an LNQD sequence, which can be proved easily as the proof of Theorem 2 from Shao [16], thus we have Observe that with n 0 = 0 Similarly, ∞ k=1 n k P |ξ 1 | > n 1/2 k < ∞.
Putting the above inequalities together yields

The LIL for linear processes generated by LNQD sequence
In this section, we will discuss the law of iterated logarithm (LIL, for short) for linear processes generated by LNQD sequence with finite variance. The linear processes are of special importance in time series analysis and they arise in wide variety of concepts (see, e.g., Hannan [20], Chapter 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. For the linear processes, Fakhre-Zakeri and Farshidi [21] established CLT under the i.i.d. assumptions and Fakhre-Zakeri and Lee [22] proved a FCLT under the strong mixing conditions. Kim and Baek [23] obtained a central limit theorem for stationary linear processes generated by linearly positively quadrant dependent process. Peligrad and Utev [24] established the central limit theorem for linear processes with dependent innovations including martingales and mixingale. Qiu and Lin [25] discussed the functional central limit theorem for linear processes with strong near-epoch dependent innovations. Dedecker et al. [26] provided the invariance principles for linear processes generated by dependent innovations. We will prove the following theorem.
The proof of Theorem 3.1 is based on the following lemmas.

Conclusions
In this paper, using the Kolmogorov type maximal inequality and Stein's method, the law of the iterated logarithm for LNQD sequence is established with less restriction of moment conditions, this improves the results of Choi [11] from E|ξ 1 | 2+δ < ∞ to E|ξ 1 | 2 < ∞. We also prove the law of the iterated logarithm for a linear process generated by LNQD sequence with the coefficients satisfying ∞ i=-∞ |a i | < ∞ by the Beveridge and Nelson decomposition, this extends the law of iterated logarithm for a linear process with the innovations from i.i.d. and NA cases to LNQD random variables.