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The improved results in almost sure central limit theorem for the maxima of strongly dependent stationary Gaussian vector sequences
Journal of Inequalities and Applications volume 2015, Article number: 224 (2015)
Abstract
The almost sure central limit theorems for the maxima of strongly dependent stationary Gaussian vector sequences are proved under some mild conditions. The results extend the ASCLT to stationary Gaussian vector sequences and give substantial improvements for the weight sequence obtained by Lin (Electron. Commun. Probab. 14:224-231, 2009).
1 Introduction and main results
In past decades, the almost sure central limit theorem (ASCLT) has been studied for independent and dependent random variables more and more profoundly. Cheng et al. [1], Fahrner and Stadtmüller [2] and Berkes and Csáki [3] considered the ASCLT for the maximum of i.i.d. random variables. For more related works on ASCLT, see [4–12]. An influential work is Csáki and Gonchigdanzan [13], which proved the following almost sure limit theorem for the maximum of a stationary weakly dependent sequence.
Theorem A
Let \(\{X_{n}:n\geq1\}\) be a standardized stationary Gaussian sequence with \(r_{n}=\operatorname{Cov}(X_{1}, X_{n+1})\) satisfying \(r_{n}\ln n(\ln\ln n)^{1+\varepsilon}=O(1)\) for some constant \(\varepsilon>0\), as \(n\rightarrow\infty\). Let \(M_{k}=\max_{i\leq k}X_{i}\). If
then
where I denotes an indicator function. Furthermore, Chen and Lin [14] extended it to the non-stationary Gaussian sequences. Chen et al. [15] extended the results to the multivariate stationary case.
Lin [16] considered the following theorem which is ASCLT version of the theorem proved by Leadbetter et al. [17].
Theorem B
Let \(\{X_{n}:n\geq1\}\) be a sequence of stationary standard Gaussian random variables with covariances \(r_{n}=\operatorname{Cov}(X_{1}, X_{n+1})\) satisfying \(|r_{n}-\frac{r}{\ln n}|\ln n(\ln\ln n)^{1+\varepsilon}=O(1)\). Let \(M_{n}=\max_{i\leq n}X_{i}\), then
where \(a_{n}\), \(b_{n}\) are defined by (1.1) and Ï• is the standard normal density function.
If \(r=0\), then (1.3) becomes (1.2). Thus Theorem A is a special case of Theorem B. The purpose of this paper is to give substantial improvements for both weight sequence and the range of random variables of Theorem B.
Throughout the paper, let \(\{\boldsymbol {Z}_{i}=(Z_{i}(1),Z_{i}(2), \ldots,Z_{i}(d)):i\geq1\}\) be a standardized stationary Gaussian vector sequence with
We write \(\boldsymbol {M} _{n}=(M_{n}(1), M_{n}(2), \ldots, M_{n}(d))\) and \(M_{n}(p)=\max_{1\leq i\leq n}Z_{i}(p)\) and shall always take \(1\leq p\neq q\leq d\); \(\boldsymbol {u}_{n}= (u_{n}(1), u_{n}(2), \ldots,u_{n}(d))\) will be a real vector, and \(\boldsymbol {u}_{n}>\boldsymbol {u}_{k}\) means \(u_{n}(p)>u_{k}(p)\) for \(p= 1, 2, \ldots, d\). For some \(\varepsilon>0\), suppose
\(\{\boldsymbol {Z}_{n}:n\geq1\}\) is called dependent: weakly dependent for \(r =0\) and strongly dependent for \(r>0\). Let \(n=|j-i|\),
r is defined by (1.4). In the paper, a very natural and mild assumption is
where
We mainly consider the ASCLT of the maximum of stationary Gaussian vector sequence satisfying (1.4), under the mild condition (1.6), which is crucial to consider other versions of the ASCLT such as that of the maximum of non-stationary strongly dependent sequence and the function of the maximum. In the sequel, \(a\ll b\) stands for \(a=O(b)\). We also define the normalized real vectors \(\boldsymbol {a}_{n}= (a_{n}, a_{n},\ldots, a_{n})\) and \(\boldsymbol {b}_{n}= (b_{n}, b_{n},\ldots, b_{n})\), where \(a_{n}\) and \(b_{n}\) are defined by (1.1). The main results are as follows.
Theorem 1
Let \(\{\boldsymbol {Z}_{n}:n\geq1\}\) be a standardized stationary Gaussian vector sequence with covariances satisfying (1.6) and \(\max_{p\neq q} (\sup_{n\geq0}|r_{n}(p,q)| )<1\). Suppose \(0\leq \alpha<1/2\), then
for \(\boldsymbol {x}=(x(1), x(2), \ldots, x(d))\in\mathbb{R}^{d}\), where \(\Phi(z)\) denotes the distribution function of a standard normal random variable.
By the terminology of summation procedures, we have the following corollary.
Corollary 1
Equation (1.8) remains valid if we replace the weight sequence \(\{d_{k}:k\geq1\}\) by \(\{d_{k}^{\ast}:k\geq1\}\) such that \(0\leq d_{k}^{\ast}\leq d_{k}\), \(\sum_{k=1}^{\infty}d_{k}^{\ast}=\infty\).
Remark 1
Our results give substantial improvements for the weight sequence in Theorem B.
Remark 2
We extend Theorem B to the stationary Gaussian vector sequences under some regularity conditions.
Remark 3
If \(\{\boldsymbol {Z}_{n}:n\geq1\}\) is a standardized stationary Gaussian sequence and \(\alpha=0\), then (1.8) becomes (1.3). Thus Theorem B is a special case of Theorem 1.
Remark 4
Essentially, the problem whether Theorem 1 holds also for some \(1/2\leq\alpha<1\) remains open.
2 Auxiliary lemmas
In this section, we present and prove some lemmas which are useful in our proof of the main result.
Lemma 1
Let \(\{\boldsymbol {Z}_{n}:n\geq1\}\) and \(\{\boldsymbol {Z}^{\prime}_{n}:n\geq1\}\) be two d-dimensional independent standardized stationary Gaussian sequences with
and
Write
Assume that (1.6) holds. Let \(\boldsymbol {u}_{i} =(u_{i}(1),u_{i}(2),\ldots,u_{i}(d))\) for \(i\geq1\) be real vectors such that \(n(1-\Phi(u_{n}(p)))\) is bounded where Φ is the standard normal distribution function. If
then
where \(K_{1}\), \(K_{2}\) are absolute constants.
Proof
See Lemma 3.1 of [15], we get the desired result. □
Lemma 2
Let \(\{Z_{n}:n\geq1\}\) be a standardized stationary Gaussian vector sequence such that condition (1.6) holds, and further suppose that \(n(1 -\Phi (u_{n}(p)))\) is bounded for \(p=1, 2, \ldots, d\) and \(\max_{p\neq q} (\sup_{n\geq0}|r_{n}(p,q)| )<1\). Then, for some \(\varepsilon>0\),
and
where \(\omega_{j}=\max\{|r_{j}(p)|,\rho_{n}\}\), \(\omega_{j}^{\prime}=\max\{|r_{ij}(p,q)|,\rho_{n}\}\).
Proof
Using (1.6) and Lemma 2.1 in [17], we get the desired result. □
Lemma 3
Let \(\{\tilde{\boldsymbol {Z}}_{n}:n\geq1\}\) be a standard stationary Gaussian vector sequence with constant covariance \(\rho_{n}(p)=r/\ln n\) for \(p=1, 2, \ldots, d\) and \(\{\boldsymbol {Z}_{n}:n\geq1\}\) satisfy the conditions of Theorem 1. Denote \(\tilde{\boldsymbol {M}}_{n}=\max_{i\leq n}\tilde{\boldsymbol {Z}}_{i}\) and \(\boldsymbol {M}_{n}=\max_{i\leq n}\boldsymbol {Z}_{i}\). Assume that \(n(1 -\Phi (u_{n}(p)))\) is bounded for \(p=1, 2, \ldots, d\) and (1.6) is satisfied. Then
Proof
Using Lemmas 1 and 2, the proof can be gained simply. □
Lemma 4
Let \(\{\boldsymbol {Z}_{n}:n\geq1\}\) be a standardized stationary Gaussian vector sequence with covariances satisfying (1.6). Suppose that the assumptions of Lemma 1 hold. Then
where \(\boldsymbol {x}=(x(1), x(2), \ldots, x(d))\in\mathbb{R}^{d}\).
Proof
Let \(\{Z_{1}^{\prime}(p), Z_{2}^{\prime}(p), \ldots, Z_{n}^{\prime}(p)\}\) have the same distribution as \(\{Z_{1}(p), Z_{2}(p), \ldots, Z_{n}(p)\}\) for \(p=1, 2, \ldots, d\), but \(\{Z_{1}^{\prime}(p), Z_{2}^{\prime}(p), \ldots, Z_{n}^{\prime }(p)\}\) is independent of \(\{Z_{1}^{\prime}(q), Z_{2}^{\prime}(q), \ldots, Z_{n}^{\prime}(q)\}\), as \(p\neq q\). Further, \(M_{n}(p)=\max_{1\leq i\leq n}Z_{i}(p)\) and \(M^{\prime}_{n}(p)=\max_{1\leq i\leq n}Z^{\prime}_{i}(p)\). By Lemma 1, we have
where \(u_{n}(p)=x(p)/a_{n}-b_{n}\).
Since \(\{Z_{1}^{\prime}(p), Z_{2}^{\prime}(p), \ldots, Z_{n}^{\prime}(p)\}\) has the same distribution as \(\{Z_{1}(p), Z_{2}(p), \ldots, Z_{n}(p)\}\), which implies \(r_{ij}^{0}(p)=r_{ij}^{\prime}(p)\). Therefore, \(A_{1}=0\).
Notice that \(\{Z_{1}^{\prime}(p), Z_{2}^{\prime}(p), \ldots, Z_{n}^{\prime}(p)\}\) is independent of \(\{Z_{1}^{\prime}(q), Z_{2}^{\prime}(q), \ldots, Z_{n}^{\prime}(q)\}\), as \(p\neq q\), thus \(r_{ij}^{\prime}(p,q)=0\). Using Lemma 3.2 in [15], we have
By (2.5), we get
From Theorem 6.5.1 of [17], we obtain
Combining this with (2.6) and (2.7), the proof is completed. □
Lemma 5
Let \(\zeta_{1}, \zeta_{2},\ldots,\zeta _{n}, \ldots\) , be a sequence of bounded random variables. If
then
Proof
The proof can be found in Lemma 2.2 obtained by Wu and Chen [18]. □
3 Proof of the main result
Proof of Theorem 1
Let \(u_{n}(p)=x(p)/a_{n}+b_{n}\) satisfy \(n(1-\Phi(u_{n}(p)))\rightarrow\tau_{p}\) for \(x(p)\in\mathbb{R}\), \(0\leq\tau_{p}<\infty\) and \(p=1,2,\ldots,d\). By Lemma 4 and the Toeplitz lemma, note that (1.8) is equivalent to
From Lemma 4, in order to prove (3.1), it suffices to prove that
Let \(\boldsymbol {\lambda}, \boldsymbol {\lambda}_{1}, \boldsymbol {\lambda}_{2}, \ldots\) be a d-dimensional independent standardized stationary Gaussian sequence with \(\boldsymbol {\lambda}, \boldsymbol {\lambda}_{1}, \boldsymbol {\lambda}_{2}, \ldots\) and \(\{\boldsymbol {Z}_{k}:k\geq1\}\) are independent. Obviously \((1-\rho_{k})^{1/2}\boldsymbol {\lambda}_{1} + \rho_{k}^{1/2}\boldsymbol {\lambda}, (1-\rho_{k})^{1/2}\boldsymbol {\lambda}_{2} + \rho_{k}^{1/2}\boldsymbol {\lambda}, \ldots\) have constant covariance \(\rho_{k}=\frac{r}{\ln k} \). Define
Obviously, \(\{\boldsymbol {M}_{k}(\rho_{k}):k\geq1\}\) and \(\{\boldsymbol {M}_{k}:k\geq1\}\) are independent.
Using the well-known \(c_{r}\)-inequality, the left-hand side of (3.2) can be written as
We will show \(L_{i}\ll \frac{D_{n}^{2}}{(\ln D_{n})^{1+\varepsilon}}\), \(i=1,2\). Let \(\boldsymbol {z}=(z(1), z(2), \ldots, z(d))\) be a real vector. Clearly,
where \(\eta_{k}=I(\boldsymbol {M}_{k}(0)\leq(1-\rho_{k})^{-1/2}(\boldsymbol {u}_{k} -\rho_{k}^{1/2}\boldsymbol {z})) -\mathbb{P}(\boldsymbol {M}_{k}(0)\leq(1-\rho_{k})^{-1/2}(\boldsymbol {u}_{k}-\rho _{k}^{1/2}\boldsymbol {z}))\).
Write
Noting that \(|\eta_{k}|\leq1\), \(\exp(\ln^{\alpha}x)=\exp (\int^{x}_{1}\frac{\alpha(\ln u)^{\alpha-1}}{u}\,\mathrm{d}u )\), we have that \(\exp(\ln^{\alpha}x)\) (\(\alpha<1/2\)) is a slowly varying function at infinity. Hence,
Using the inequality \(x^{n-i}-x^{n}\leq\frac{i}{n}\) for \(0< x<1\), \(i\leq n\), we get
So, we have
For \(T_{1}\), we have
According to Wu and Chen [18], for sufficiently large n, for \(\alpha>0\), we have
Since \(\alpha<1/2\) implies \((1-\alpha)/\alpha>1\), letting \(0<\varepsilon<(1-\alpha)/\alpha-1\), for sufficiently large n, we get
Combining with (3.7)-(3.10), we can get
By (3.5), (3.6) and (3.11), we have
Clearly,
Similarly to (3.6), we find that \(J_{1}\leq\sum^{\infty}_{k=1}d_{k}^{2}<\infty\). Note that
For \(J_{21}\), we can get
By Lemma 3 and (3.9), for \(\alpha>0\), we have
Let \(\boldsymbol {z}=(z(1), z(2), \ldots, z(d))\) be a real vector. By (3.7)-(3.11), we obtain
For \(J_{22}\), noting that \(\{\boldsymbol {M}_{i}:i\geq1\}\) and \(\{\boldsymbol {M}_{i}(\rho_{i}):i\geq1\}\) are independent, by Lemma 3 and (3.16), we get
By (3.17), we have
Together with (3.20) and (3.21), we obtain
Hence, by (3.19) and (3.22), we have
By (3.13), (3.14), (3.18) and (3.23), for \(\alpha>0\), we get
Thus (3.2)-(3.24) together establish (3.18). The proof is completed. □
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Acknowledgements
The authors were very grateful to the editor and two anonymous referees for their careful reading of the manuscript and valuable suggestions which helped in significantly improving an earlier version of this article. Supported by the National Natural Science Foundation of China (11361019), project supported by Program of the Guangxi China Science Foundation (2013GXNSFDA019001, 2014GXNSFAA118015, 2013GXNSFAA278003), and the Support of the Scientific Research Project of Education Department of Guangxi (YB2014150).
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XZ conceived of the study and drafted, completed the manuscript. QW participated in the discussion of the manuscript. XZ and QW read and approved the final manuscript.
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Xiang Zeng is a lecturer, Master, working in the field of probability and statistics.
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Zeng, X., Wu, Q. The improved results in almost sure central limit theorem for the maxima of strongly dependent stationary Gaussian vector sequences. J Inequal Appl 2015, 224 (2015). https://doi.org/10.1186/s13660-015-0740-2
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DOI: https://doi.org/10.1186/s13660-015-0740-2