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The law of the iterated logarithm for LNQD sequences
Journal of Inequalities and Applications volume 2018, Article number: 11 (2018)
Abstract
Let \(\{\xi_{i},i\in{\mathbb{Z}}\}\) 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 \(\sum_{i=-\infty}^{\infty}|a_{i}|<\infty\) by a Beveridge and Nelson decomposition.
1 Introduction
Two random variables X and Y are said to be negatively quadrant dependent (NQD, for short), if \(P(X\leq x,Y\leq y)-P(X\leq x)P(Y\leq y)\leq0\) for all \(x,y\in {\mathbb{R}}\). A sequence \(\{X_{k},k\in{\mathbb{Z}}\}\) is said to be linear negatively quadrant dependent (LNQD, for short) if for any disjoint finite subsets \(A,B\subset{\mathbb{Z}}\) and any positive real numbers \(r_{j}\), \(\sum_{i\in A}r_{i}X_{i}\) and \(\sum_{j\in B}r_{j}X_{j}\) 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. \(\{X_{i},i\in I\}\) 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 \(G: {\mathbb{R}}^{A}\rightarrow{\mathbb{R}}\) and \(H: {\mathbb {R}}^{B}\rightarrow{\mathbb{R}}\) and \(EG^{2}(X_{i},i\in A)<\infty\), \(EH^{2}(X_{j},j\in B)<\infty\), we have \(\operatorname{Cov} (G(X_{i},i\in A),H(X_{j},j\in B))\leq0\). 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 \(\log x=\ln{\max\{e,x\}}\).
2 Main results
We will need the following property.
-
(H1)
(Hoeffding equality): For any absolutely continuous functions f and g on \({\mathbb{R}}^{1}\) and for any random variables X and Y satisfying \(Ef^{2}(X)+Eg^{2}(Y)<\infty\), we have
$$\begin{aligned}[b] &\operatorname{Cov}\bigl(f(X),g(Y)\bigr)\\ &\quad= \int_{-\infty}^{\infty} \int_{-\infty }^{\infty}f'(x)g'(y) \bigl\{ P(X\geq x,Y\geq y)-P(X\geq x)P(Y\geq y)\bigr\} \,dx\,dy.\end{aligned} $$
Now we state the law of iterated logarithm for LNQD sequence.
Theorem 2.1
Let \(\{\xi_{i},i\geq1\}\) be a strictly stationary LNQD sequence with \(E\xi_{i}=0\), \(E\xi_{i}^{2}<\infty\) and \(\sigma^{2}=E\xi_{1}^{2}+2\sum_{i=2}^{\infty}E\xi_{1}\xi_{i}>0\). Put \(S_{n}=\sum_{i=1}^{n}\xi_{i}\). Then
Remark 2.2
Our theorem extends the corresponding results of Corollary 1.2 in Choi [11]. Choi established a law of the iterated logarithm for LNQD sequence with \(E|\xi_{1}|^{2+\delta}<\infty\) for some \(\delta>0\) and variance coefficients decaying polynomially. But our theorem only restricts the finite variance.
The proof of Theorem 2.1 is based on the following lemmas.
Lemma 2.3
(Lehmann [1])
Let random variables X and Y be NQD, then
-
1.
\(EXY\leq EXEY\);
-
2.
\(P(X >x, Y> y) \leq P(X >x)P(Y> y)\);
-
3.
if f and g are both nondecreasing (or both nonincreasing) functions, then \(f(X)\) and \(g(Y)\) are NQD.
Lemma 2.4
Let \(\{\xi_{i}, 1\leq1\leq n\}\) be an LNQD sequence of random variables with mean zero and finite second moments. Let \(S_{n}=\sum_{i=1}^{n}\xi_{i}\) and \(B_{n}=\sum_{i=1}^{n}E\xi_{i}^{2}\). Then, for all \(x>0\), \(a>0\) and \(0<\alpha<1\), we know
In particular, we have
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 \(\{Y_{i}, 1\leq i\leq n\}\) be an LNQD sequence of random variables with \(EY_{i}=0\) and \(E|Y_{i}|^{3}<\infty\). Define \(T_{n}=\sum_{i=1}^{n}Y_{i}\) and \(B_{n}^{2}=\sum_{i=1}^{n}EY_{i}^{2}\). Then, for any \(x>0\),
where Φ 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])
Let \(\zeta_{i}=Y_{i}/{B_{n}}\), \(W=\sum_{i=1}^{n}\zeta_{i}\), \(W^{(i)}=W-\zeta_{i}\),
Obviously, \(\zeta_{i}=\zeta_{i,1}+\zeta_{i,2}\). Write
where
By the definition of LNQD, we know \(\zeta_{i}\) and \(W^{(i)}\) are NQD, then by (H1) and (2.6) we have
By (2.6), we obtain
To estimate \(R_{4}\), let \(K_{i}(t)=E(\zeta_{i,1}I\{0\leq t\leq\zeta_{i,1}\} -\zeta_{i,1}I\{\zeta_{i,1}\leq t< 0\})\). Rewrite \(R_{4}\) as
For fixed \(0< t<1\), \(xI\{0\leq t\leq x\}\) is a nondecreasing functions of x, by the definition of LNQD and Lemma 2.3, \(\zeta_{i,1}I\{ 0\leq t\leq\zeta_{i,1}\}\) and \(W^{(i)}\) are NQD. Then by (H1) and (2.7),
Similarly,
Let \(R_{5}=|R_{1}|+|R_{2}|+|R_{3}|+|R_{4,2}|+|R_{4,3}|\). Observe that
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 \(0<\varepsilon<\frac{1}{30}\)
and
Let m be an integer such that
Put \(a_{i}=\varepsilon\sigma(i/\log\log i)^{1/2}/m\). Define
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
Observe that
for every n sufficiently large,
and
Hence, by (2.10)
provided that n is sufficiently large.
By the definition of LNQD and Lemma 2.3, we know \(\{u_{i},i\geq1\} \) are also LNQD random variables with \(Eu_{i}=0\) and \(|u_{i}|\leq2ma_{im}\) for every i. By Lemma 2.4 (with \(\alpha =1-\varepsilon\), \(a=2ma_{n}\)), (2.13) and (2.14), we get
for every sufficiently large n. By using the standard subsequence method, (2.15) yields
Now (2.8) follows by (2.12) and (2.16).
To prove (2.9), let
It suffices to show that
In fact, by Lemma 2.4, similar to the proof of (2.15), we obtain
Then by (2.17), we have
By the definition of LNQD and Lemma 2.3, we see that \(\{S_{n_{k},1}-S_{n_{k-1},1}, k\geq1\}\) is an LNQD sequence, then, for any \(x>0\), \(y>0\), \(k\neq j\),
Hence, by the generalized Borel-Cantelli lemma (see, e.g., Kochen and Stone [19]), (2.18) yields
which together with (2.8) and (2.12) gives
and hence (2.9) holds.
To verify (2.17), set
Obviously, \(S_{n_{k},1}=T_{k,1}+T_{k,2}\). Then by Lemma 2.4, similar to the proof of (2.15), we obtain
Thus, we only need to show that
It is easy to see that
From Lemma 2.5, we obtain
where
Obviously, we have
Noting that \(\{v_{i,1},1\leq i\leq k^{4}\}\) 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,
Putting the above inequalities together yields
This proves (2.19), by combining the above inequalities (2.20)-(2.23). □
3 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.
Theorem 3.1
Let \(\{\xi_{i},i\in\mathbb{Z}\}\) be a strictly stationary LNQD sequence with \(E\xi_{i}=0\), \(E\xi_{i}^{2}<\infty\) and \(\sigma^{2}=E\xi_{1}^{2}+2\sum_{i=2}^{\infty}E\xi_{1}\xi_{i}>0\). \(\{a_{j},j\in \mathbb{Z}\}\) be a sequence of real numbers with \(\sum_{j=-\infty}^{\infty}|a_{j}|<\infty\). Define the linear processes \(X_{t}=\sum_{i=-\infty}^{\infty}a_{i}\xi_{t-i}\). Then
The proof of Theorem 3.1 is based on the following lemmas.
Lemma 3.2
Let \(\{\xi_{i},i\in\mathbb{Z}\}\) be a strictly sequence of random variables, \(\{a_{n},n\geq1\}\) be a monotone decreasing sequence of nonnegative real numbers. Then \(\forall j\in\mathbb{Z}\),
Proof
Let \(Y_{j}=\sup_{n\geq1}|a_{n}\sum_{i=1}^{n}\xi_{i-j}|\), \(Y=\sup_{n\geq 1}|a_{n}\sum_{i=1}^{n}\xi_{i}|\). Obviously
similarly,
By the strictly stationarity, we know \((\xi_{1-j},\xi_{2-j},\ldots,\xi_{t-j})\stackrel{\mathrm{d}}{=}(\xi _{1},\xi_{2},\ldots,\xi_{t})\), then, for every Borel set \(D\in\mathbb{R}^{t}\),
In particular, if we take \(D= \{(x_{1},x_{2},\ldots,x_{n}):\max_{1\leq t\leq k}|a_{t}\sum_{i=1}^{t}\xi_{i}|\leq x \}\), then the result of Lemma 3.2 can be obtained by the above statements. □
Lemma 3.3
Let \(\{\xi_{i}, i\in \mathbb{Z}\}\) be a strictly stationary LNQD sequence of random variables with \(E\xi_{1}=0\), \(E{\xi_{1}}^{2}<\infty\), \({\sigma}^{2}=E{\xi_{1}}^{2}+2\sum_{i=2}^{\infty} E\xi_{1}\xi_{i}>0\). Then
Proof
Let \(b_{n}=(2n\log\log{n})^{\frac {1}{2}}\), \({b_{2^{k}}}/{b_{2^{k+1}}}\rightarrow \frac{\sqrt{2}}{2}\) (\(k\rightarrow \infty\)), then there exists \(C_{1}>0\), such that for all \(k\geq 0\), \({b_{2^{k}}}/{b_{2^{k+1}}}\geq C_{1}\). Let m, \(\sigma_{m}^{2}\), \(a_{i}\), \(g_{l}(a_{i},\xi_{i})\), \(Y_{i,l}\), \(S_{i,l}\), \(u_{i}\), \(U_{i}\) be defined as in the proof of Theorem 2.1. Note that \(\sum_{k=1}^{n}\xi_{k}=S_{n,1}+S_{n,2}\). Then
In order to prove (3.2), it is sufficient to prove
Note that
The last inequalities can be induced by the same argument as in (2.11).
Finally, in order to prove (3.2), it remains to check that (3.4) holds by combining the above inequalities. We have
where B will be given later. Noting the choice of \(C_{1}\), we have
It is easy to check that
By the same argument as of (2.14), there exists \(k_{0}\), such that, for every \(k\geq k_{0}\),
By the definition of LNQD and Lemma 2.3, we know \(\{u_{i},i\geq1\} \) are also LNQD random variables with \(Eu_{i}=0\) and \(|u_{i}|\leq2ma_{im}\) for every i. By Lemma 2.4 (with \(\alpha =1-\varepsilon\), \(a=2ma_{2^{k+1}}\)), then by (3.9) and (3.10), observing that \(0<\varepsilon<\frac{\sqrt{2}C_{1}}{2\sigma}\), we have
where
and choose B sufficiently large such that \(BD>1\). Thus (3.4) holds by combining the above inequalities together. □
Proof of Theorem 3.1
By a Beveridge and Nelson decomposition for a linear process, for \(m,n,t\in\mathbb{N}\), let
Obviously
By the strictly stationarity, for every \(\varepsilon>0\), we have
Then by the Borel-Cantelli lemma, for any \(j\geq0\),
Therefore
Similarly, we obtain
By the above statement, we have
By Theorem 2.1
From the definition of LNQD and Lemma 2.3, it is easy to check that \(\{-\xi_{i};i\in\mathbb{Z}\}\) is an LNQD sequence of random variables. Then, by Theorem 2.1,
Thus
Let \(S_{n}=\sum_{t=1}^{n}X_{t}\), combining (3.12)-(3.16), then
Then by the strictly stationarity, Lemma 3.2 and Lemma 3.3, we know
Then, by (3.18),
By (3.19), letting \(m\to\infty\) in (3.17), we have
On the other hand, by (3.13), (3.15) and (3.16), we obtain
Then, letting \(m\to\infty\),
Hence from (3.20) and (3.22) the desired conclusion (3.1) follows. □
4 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 \(E|\xi _{1}|^{2+\delta}<\infty\) to \(E|\xi_{1}|^{2}<\infty\). We also prove the law of the iterated logarithm for a linear process generated by LNQD sequence with the coefficients satisfying \(\sum_{i=-\infty}^{\infty}|a_{i}|<\infty\) 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.
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Acknowledgements
The author greatly appreciates both the editors and the referees for their valuable comments and some helpful suggestions that improved the clarity and readability of this paper. The paper is supported by NSFC (Grant No. 11771178); the Science and Technology Development Program of Jilin Province (Grant No. 20170101152JC).
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Zhang, Y. The law of the iterated logarithm for LNQD sequences. J Inequal Appl 2018, 11 (2018). https://doi.org/10.1186/s13660-017-1607-5
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DOI: https://doi.org/10.1186/s13660-017-1607-5
Keywords
- law of the iterated logarithm
- linear process
- Stein’s method
- LNQD sequence
- Beveridge and Nelson decomposition