Open Access Research

Asymptotic properties of wavelet-based estimator in nonparametric regression model with weakly dependent processes

Xing-cai Zhou12*, Jin-guan Lin1 and Chang-Ming Yin3

Author Affiliations

1 Department of Mathematics, Southeast University, Nanjing, 210096, P.R. China

2 Department of Mathematics and Computer Science, Tongling University, Tongling, Anhui, 244000, P.R. China

3 School of Mathematics and Information Science, Guangxi University, Guangxi, 530004, P.R. China

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Journal of Inequalities and Applications 2013, 2013:261  doi:10.1186/1029-242X-2013-261


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/261


Received:2 April 2013
Accepted:8 May 2013
Published:23 May 2013

© 2013 Zhou et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider a nonparametric regression model with replicated observations based on the φ-mixing and the ρ-mixing error’s structures respectively, for exhibiting dependence among the units. The wavelet procedures are developed to estimate the regression function. Under suitable conditions, we obtain expansions for the bias and the variance of wavelet estimator, prove the moment consistency, the strong consistency, the strong convergence rate of it, and establish the asymptotic normality of wavelet estimator.

Keywords:
nonparametric regression model; wavelet; weak dependence; φ-mixing; ρ-mixing

1 Introduction

Consider the following nonparametric regression model:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M1">View MathML</a>

(1.1)

from a discrete set of observations of the process <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M2">View MathML</a> at the points <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M3">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M4">View MathML</a> is a zero mean stochastic process, defined on a probability space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M5">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M6">View MathML</a> is an unknown function defined on a closed interval <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M7">View MathML</a>.

It is well known that regression model has a wide range of applications in filtering and prediction in communications and control systems, pattern recognition, classification and econometrics, and is an important tool of data analysis. Of much interest about the problem have been the weighted function estimates of g; see, for example, Priestley and Chao [1], Gasser and Müller [2,3], Prakasa Rao [4], Clark [5] and the references therein for the independent case; Roussas [6], Fan [7], Roussas and Tran [8], Liang and Jing [9], Yang and Li [10], Yang [11] for the various dependent cases.

In this article, we discuss a nonparametric estimation problem in the model (1.1) with repeated measurements. We assume that a random sample of m experimental units is available and the observed data for the jth unit are the values, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M8">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M9">View MathML</a>), of a response variable corresponding to the values, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M10">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M9">View MathML</a>), of a controlled variable. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M12">View MathML</a> obey the model (1.1), i.e.,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M13">View MathML</a>

(1.2)

where the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M10">View MathML</a>’s are fixed with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M15">View MathML</a>, and the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M16">View MathML</a>’s are zero mean random errors. The model can be applied to many fields. For instance, in hydrology many phenomena may be represented by a sequence of continuous responses <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M17">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M10">View MathML</a> represents the time elapsed from the beginning of a certain year, j indicates the corresponding year, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M16">View MathML</a> is the measurement of the deviation from the annual mean <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M20">View MathML</a>; in some biological phenomena as the growth of individual (or populations) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M17">View MathML</a> will be the growth points of the j-individual, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M16">View MathML</a> the measurement of the deviation from the mean growth <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M20">View MathML</a> of the response of the j-individual, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M24">View MathML</a> the points where measurements are taken [12]. It is clear that the observations <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M8">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M26">View MathML</a>) made on the same experimental unit will in general be correlated. Hart and Wehrly [13] studied the asymptotic square mean error of a kernel estimator in the model with zero mean errors <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M16">View MathML</a>’s be correlated, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M28">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M29">View MathML</a> and zero elsewhere where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M30">View MathML</a> is a correlation function. The uncorrelated assumption between units is unrealistic. In practice, sometimes the observed responses in different units are also correlated, more precisely a sequence of responses curve <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M31">View MathML</a> has an intrinsic dependence structure, such as mixing conditions. Under a weak error’s structure among units, Fraiman and Iribarren [12] proposed nonparametric estimates of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32">View MathML</a> in model (1.2) based on locally weighted averages, and gave their local and global asymptotic behaviors.

In the paper, we develop wavelet methods to estimate a regression function in the model (1.2) with the φ-mixing and ρ-mixing error’s structures respectively, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M33">View MathML</a> is a φ-mixing or ρ-mixing process. One motivation for using wavelet for the model (1.2) is that most of the nonparametric analyses of regression models are based on an important assumption that the regression function is smooth; but in reality, the assumption may not be satisfied. The major advantage of the wavelet is that the hypotheses of degrees of smoothness of the underlying function are less restrictive. Up to now, there have been no results on wavelet estimation for model (1.2). Another motivation for considering the model (1.2) with weakly dependent processes is that our interest is to avoid as far as possible any assumptions on the error’s structure within <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M34">View MathML</a> for each j (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M35">View MathML</a>), and in the meantime to exhibit weakly dependence among the units.

For a systematic discussion of wavelets and their applications in statistics, see the recent monographs by Härdle et al.[14] and Vidakovic [15]. Due to their ability to adapt to local features of unknown curves, many authors have applied wavelet procedures to estimate the general nonparametric model. See recent works, for example, Antoniadis et al.[16] and Xue [17] on independent errors; Johnstone and Silverman [18] for correlated noise; Liang et al.[19] on martingale difference errors; Li and Xiao [20] for long memory data; Li et al.[21] on associated samples; Xue [22], Sun and Chai [23], Li and Guo [24] and Liang [25] on mixing error assumptions.

For dealing with weakly dependent data, bootstrap and blockwise are well known. They are useful techniques of resampling, which can preserve the dependent properties of the data by appropriately choosing blocks of data. They have been sufficiently investigated by many papers, for example, Bühlman and Künsch [26], Yuichi [27], Lahiri [28], Lin and Zhang [29,30] and Lin et al.[31]. For the nonparametric regression model without repeated observations under weakly dependent processes, Lin and Zhang [30] respectively adopted bootstrap wavelet and blockwise bootstrap wavelet to generate an independent blockwise sample from the original dependent data, defined the wavelet estimators of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32">View MathML</a>, and then took advantage of the independence of the blockwise sample to prove some asymptotic properties of the wavelet estimators. In addition, the weakly dependent conditions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M37">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M38">View MathML</a> in Lin and Zhang [30] are weaker than the conditions for the consistency and asymptotic normality of ρ-mixing and φ-mixing dependent sequences. In our paper, we consider the nonparametric regression model with repeated observations under the specific ρ-mixing and φ-mixing dependent processes. We do not use bootstrap and/or blockwise to define wavelet estimator of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32">View MathML</a>, but construct it by the following simple formula (2.2), and can show directly some asymptotic behaviors of the wavelet estimator by applying the fundamental properties of ρ-mixing and φ-mixing sequences and some proof techniques.

Recall the definitions of the sequences of the φ-mixing and ρ-mixing random variables. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M40">View MathML</a> be a sequence of random variables defined on a probability space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M41">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M42">View MathML</a> be σ-algebra generated by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M43">View MathML</a>, and denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M44">View MathML</a> to be the set of all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M45">View MathML</a>-measurable random variables with second moments.

A sequence of random variables <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M40">View MathML</a> is said to be φ-mixing if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M47">View MathML</a>

A sequence of random variables <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M40">View MathML</a> is said to be ρ-mixing if maximal correlation coefficient

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M49">View MathML</a>

The concept of a mixing sequence is central in many areas of economics, finance and other sciences. A mixing time series can be viewed as a sequence of random variables for which the past and distant future are asymptotically independent. A number of limit theorems for φ-mixing and ρ-mixing random variables have been studied by many authors. For example, see Shao [32], Peligrad [33], Utev [34], Kiesel [35], Chen et al.[36] and Zhou [37] for φ-mixing; Peligrad [38], Peligrad and Shao [39,40], Shao [41] and Bradley [42] for ρ-mixing. Some limit theories can be found in the monograph of Lin and Lu [43].

The article is structured as follows. In Section 2, we introduce the wavelet estimation procedures and establish main results. The proofs of the main results are provided in Section 3.

2 Estimators and main results

Defining <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M50">View MathML</a>, from (1.2), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M51">View MathML</a>

(2.1)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M52">View MathML</a>. Expressing the model in this way is useful since the problem of estimating <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32">View MathML</a> may now be regarded as that of fitting a curve through the sample means <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M54">View MathML</a>. The wavelet technique is applied to estimate the regression function in model (1.2). The detailed procedure is summarized below.

For convenience, we introduce some symbols and definitions along the line of Antoniadis et al.[16]. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M55">View MathML</a> is a given scaling function in the Schwartz space with order l. A multiresolution analysis of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M56">View MathML</a> consists of an increasing sequence of closed subspaces <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M57">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M58">View MathML</a> , where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M56">View MathML</a> is the set of square integrable functions over the real line. The associated integral kernel of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M57">View MathML</a> is given by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M61">View MathML</a>

where Z denotes the set of integers, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M62">View MathML</a> is an integer depending only on n. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M63">View MathML</a> be a partition of the interval ℐ with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M64">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M65">View MathML</a>. From (2.1), we now construct the wavelet estimator of g,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M66">View MathML</a>

(2.2)

In the sequel, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M67">View MathML</a> denote generic finite positive constants, whose values are unimportant and may change from line to line. Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M68">View MathML</a> and suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M69">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M70">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M71">View MathML</a>.

Before formulating the main results, we first give some assumptions, which are quite mild and can be easily satisfied.

A1.

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M72">View MathML</a> is a sequence of φ-mixing (ρ-mixing);

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M72">View MathML</a> is a sequence of identically distributed φ-mixing (ρ-mixing);

(iii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M72">View MathML</a> is a sequence of strictly stationary φ-mixing (ρ-mixing).

A2.

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M75">View MathML</a>;

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M76">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M77">View MathML</a>);

(iii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M78">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M79">View MathML</a>;

(iv) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M80">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M81">View MathML</a>) for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M82">View MathML</a>.

A3. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M83">View MathML</a>.

A4. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M84">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M85">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M86">View MathML</a> is the Sobolev space of order υ, i.e., if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M87">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M88">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M89">View MathML</a> denotes the Fourier transform of h.

A5. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M32">View MathML</a> satisfies the Lipschitz condition of order <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M91">View MathML</a>.

A6. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M92">View MathML</a> is q regular with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M93">View MathML</a>, satisfies the Lipschitz condition with order 1 and has a compact support, and satisfies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M94">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M95">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M96">View MathML</a> is the Fourier transform of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M92">View MathML</a>.

A7. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M98">View MathML</a>.

A8. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M99">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M100">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M101">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M102">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M101">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M104">View MathML</a>, such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M105">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M106">View MathML</a>, where υ and γ are defined in (A4) and (A5), respectively.

Remark 2.1 We refer to the monograph of Doukhan [44] for properties of φ-mixing and ρ-mixing, and more mixing conditions.

Remark 2.2 It is well known that (A4)-(A7) are the mild regularity conditions for wavelet smoothing; see Antoniadis et al.[16], Chai and Xu [45], Xue [22], Sun and Chai [23], Zhou and You [46] and Li and Guo [24].

Remark 2.3 (A8) is satisfied easily. For example, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M107">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M108">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M109">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M110">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M111">View MathML</a>.

Our results are listed as follows.

Theorem 2.1Assume that (A1)(i) and (A2)(i), and (A4)-(A7) are satisfied. Then

(a) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M112">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M113">View MathML</a>is defined in Lemma 3.2;

(b) φ-mixing: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M114">View MathML</a>;

ρ-mixing: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M115">View MathML</a>.

Theorem 2.2Under (A1)(i), (A2)(i)(ii), and (A4)-(A7), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M116">View MathML</a>

Theorem 2.3Assume that [(A1)(ii) and (A2)(i)(ii)] or [(A1)(i) and (A2)(iii)], and (A4)-(A7) are satisfied. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M117">View MathML</a>

Theorem 2.4Assume that (A1)(ii), (A2)(i)(ii), and (A4)-(A7) are satisfied. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M118">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M119">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M120">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M121">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M122">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M123">View MathML</a>

Theorem 2.5Assume that (A1)(iii), (A2)(i)(iv), (A3), and (A4)-(A8) are satisfied. For a fixedxand each<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M124">View MathML</a>, there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M125">View MathML</a>verifying<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M126">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M127">View MathML</a>as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M128">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M129">View MathML</a>

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M130">View MathML</a>denotes convergence in distribution.

3 Proofs of the main results

In order to prove the main results, we first present several lemmas.

Lemma 3.1Suppose that (A6) holds. We have

(a) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M131">View MathML</a>.

(b) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M132">View MathML</a>.

(c) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M133">View MathML</a>uniformly in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M134">View MathML</a>, as<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M100">View MathML</a>.

The proofs of (a) and (b), and (c) respectively can be found in Antoniadis et al.[16] and Walter [47].

Lemma 3.2Suppose that (A6)-(A7) hold, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M136">View MathML</a>satisfies (A4)-(A5). Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M137">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M138">View MathML</a>

It follows easily from Theorem 3.2 of Antoniadis et al. [16].

Lemma 3.3 (a) Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139">View MathML</a>be aφ-mixing sequence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M140">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M141">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M142">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M143">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M144">View MathML</a>

(b) Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139">View MathML</a>be aρ-mixing sequence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M140">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M147">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M148">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M143">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M150">View MathML</a>

Lemmas 3.3(a) and (b) respectively come from Lemmas 10.1.d and 10.1.c of Lin and Lu[43].

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M151">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M70">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M153">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M154">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M155">View MathML</a>. The following Lemma 3.4(a) and (b) can be found in Shao [32] and Shao [41] respectively.

Lemma 3.4 (a) Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139">View MathML</a>be aφ-mixing sequence.

(i) If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M157">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M158">View MathML</a>

(ii) Suppose that there exists an array<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M159">View MathML</a>of positive numbers such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M160">View MathML</a>for every<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M155">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M162">View MathML</a>. Then, for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M163">View MathML</a>, there exists a positive constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M164">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M165">View MathML</a>

(b) Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139">View MathML</a>be aρ-mixing sequence with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M157">View MathML</a>. Then, for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M163">View MathML</a>, there exists a positive constant<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M169">View MathML</a>such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M170">View MathML</a>

Lemma 3.5Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M171">View MathML</a>be aφ-mixing (ρ-mixing) sequence of identically distributed random variables with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M172">View MathML</a>

for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M173">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M174">View MathML</a>

Lemma 3.5 can be found in Theorem 8.2.2 of Lin and Lu [43]. Ibragimov [48,49] gave the following Lemma 3.6, which also can be found in Lin and Lu [43].

Lemma 3.6Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M139">View MathML</a>be a strictly stationaryφ-mixing (ρ-mixing) sequence of random variables with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M176">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M177">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M178">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M76">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M180">View MathML</a>), then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M181">View MathML</a>

We are now in a position to give the proofs of the main results.

Proof of Theorem 2.1 From (1.2) and (2.2), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M182">View MathML</a>

(3.1)

By Lemma 3.2, (a) holds.

Denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M183">View MathML</a>. By <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M184">View MathML</a> and Lemma 3.1(b),

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M185">View MathML</a>

(3.2)

For φ-mixing, by Lemmas 3.3(a) and (3.2), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M186">View MathML</a>

Therefore, (b) holds for φ-mixing. Similar to the arguments, we obtain (b) for ρ-mixing by Lemma 3.3(b). □

Proof of Theorem 2.2 We know that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M187">View MathML</a>. It follows easily from Theorem 2.1. □

Proof of Theorem 2.3 From (1.2) and (2.2), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M188">View MathML</a>

(3.3)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M189">View MathML</a> is defined in the proof of Theorem 2.1. Note that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M190">View MathML</a>

(3.4)

as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M101">View MathML</a> by Lemma 3.2. It remains to show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M192">View MathML</a>

(3.5)

(1) φ-mixing. Here, we consider <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M193">View MathML</a> under two different assumptions: [(A1)(ii) and (A2)(i)(ii)] and [(A1)(i) and (A2)(iii)], respectively.

If the assumptions are [(A1)(ii) and (A2)(i)(ii)], denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M194">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M195">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M196">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M197">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M198">View MathML</a>. Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M199">View MathML</a>. We have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M200">View MathML</a>

(3.6)

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M201">View MathML</a>, by Lemma 3.4(a), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M202">View MathML</a>

Therefore, it follows from the Borel-Cantelli lemma that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M203">View MathML</a>

(3.7)

Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M204">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M205">View MathML</a>). By Lemma 3.5, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M206">View MathML</a> a.s. Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M207">View MathML</a>

(3.8)

By (3.8) and Lemma 3.1(ii), one gets

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M208">View MathML</a>

(3.9)

Further, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M209">View MathML</a>

(3.10)

From (3.6), (3.7), (3.9) and (3.10), we obtain (3.5).

If the assumptions are [(A1)(i) and (A2)(iii)], note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M210">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M104">View MathML</a>, hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M212">View MathML</a>. Further, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M213">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M214">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215">View MathML</a>. Take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M216">View MathML</a>, next take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215">View MathML</a> small enough such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M218">View MathML</a>. By Lemma 3.4(a), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M219">View MathML</a>

Therefore, from the Borel-Cantelli lemma, we obtain (3.5).

(2) ρ-mixing. We also consider <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M193">View MathML</a> under two different assumptions: [(A1)(ii) and (A2)(i)(ii)] and [(A1)(i) and (A2)(iii)], respectively.

Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M221">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M104">View MathML</a>, hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M223">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M163">View MathML</a>. Further, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M225">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M214">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215">View MathML</a>.

If the assumptions are [(A1)(ii) and (A2)(i)(ii)], from (3.6)-(3.10), it is known that we only need to prove (3.7) for obtaining (3.5). Taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M201">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M229">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M230">View MathML</a>. Next, take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215">View MathML</a> small enough such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M232">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M233">View MathML</a>. By Lemma 3.4(b), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M234">View MathML</a>

Therefore, (3.7) holds.

If the assumptions are [(A1)(i) and (A2)(iii)], take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M235">View MathML</a>, next take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215">View MathML</a> small enough such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M237">View MathML</a>. By Lemma 3.4(b), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M238">View MathML</a>

Thus, we obtain (3.5).

So, we complete the proof of Theorem 2.3. □

Proof of Theorem 2.4 Here, we use some symbols of the proof of Theorem 2.3. From (3.3), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M239">View MathML</a>

(3.11)

By Lemma 3.2, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M118">View MathML</a>, one gets

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M241">View MathML</a>

(3.12)

Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M242">View MathML</a> satisfies the Lipschitz condition of order 1 on x. We have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M243">View MathML</a>

(3.13)

Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M244">View MathML</a>, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M245">View MathML</a>, by Lemma 3.5, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M246">View MathML</a>

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M247">View MathML</a>

(3.14)

By (3.14), one gets

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M248">View MathML</a>

Thus, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M249">View MathML</a>

(3.15)

By Lemma 3.1(ii), (3.14), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M250">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M251">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M252">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M253">View MathML</a>

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M254">View MathML</a>

(3.16)

To complete the proof of the theorem, it is suffices to show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M255">View MathML</a>

(3.17)

by (3.11)-(3.13) and (3.15)-(3.16).

Here, we show (3.17) under φ-mixing and ρ-mixing, respectively.

(1) φ-mixing. Taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M256">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M257">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M258">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M259">View MathML</a>. By Lemma 3.4(a), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M260">View MathML</a>

Thus, (3.17) holds by the Borel-Cantelli lemma.

(2) ρ-mixing. Similar to the arguments in the proof of Theorem 2.3, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M261">View MathML</a> for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M214">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215">View MathML</a>. Taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M256">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M257">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M258">View MathML</a>. Next, take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M215">View MathML</a> small enough such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M268">View MathML</a>. By Lemma 3.4(b), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M269">View MathML</a>

Therefore, we also obtain (3.17).

So, we complete the proof of Theorem 2.4. □

Proof of Theorem 2.5 Denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M270">View MathML</a>. For each <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M134">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M272">View MathML</a>

Since we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M273">View MathML</a>

by Lemma 3.2 and (A8), it suffices to show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M274">View MathML</a>

(3.18)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M275">View MathML</a>

(3.19)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M276">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M277">View MathML</a>. Under the assumption of φ-mixing, denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M278">View MathML</a>; if we consider ρ-mixing, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M279">View MathML</a> is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M30">View MathML</a>. By Lemma 3.3, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M281">View MathML</a>

Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M282">View MathML</a>

which can be made arbitrarily small if we first choose ς such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M283">View MathML</a> is small, and for this ς we choose m large enough so that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M102">View MathML</a> makes <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M285">View MathML</a> arbitrarily small. Thus, we obtain (3.18).

It remains to establish (3.19). Denote <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M286">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M287">View MathML</a>. By Lemma 3.4, and (iv) implies (ii), for φ-mixing and ρ-mixing, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M288">View MathML</a>

(3.20)

For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M289">View MathML</a>, from (3.20) and the dominated convergence theorem,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M290">View MathML</a>

Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M291">View MathML</a> converges absolutely for φ-mixing and ρ-mixing, that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M292">View MathML</a>

(3.21)

We easily obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M293">View MathML</a>

since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M294">View MathML</a> by Lemma 3.6, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M295">View MathML</a> by (3.21), and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M296">View MathML</a> by Lemma 3.1.

Thus, we complete the proof of Theorem 2.5. □

4 Conclusion and discussion

The paper studies a nonparametric regression model with replicated observations under weakly dependent processes by wavelet procedures. For exhibiting dependence among the units, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M297">View MathML</a> is a φ-mixing or ρ-mixing process, and avoid as far as possible any assumptions on the error’s structure among <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M34">View MathML</a> for each j (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/261/mathml/M35">View MathML</a>). Under suitable conditions, we obtain expansions for the bias and the variance of wavelet estimator, prove the moment consistency, the strong consistency, the strong convergence rate of it, and establish the asymptotic normality of wavelet estimator. For the general nonparametric model, consistency results of linear wavelet estimator can be derived from general results on regression estimators in the case of dependent errors. But our results cannot be derived directly from general results on regression estimators because the nonparametric regression model with repeated measurements we considered has complex dependent error’s structure. Bootstrap and blockwise are useful techniques of resampling, which can preserve the dependent properties of the data by appropriately choosing blocks of data. They have been sufficiently investigated for weakly dependent data by many papers, for example, Bühlman and Künsch [26], Yuichi [27], Lahiri [28], Lin and Zhang [29,30] and Lin et al.[31]. In the future, we may try bootstrap and blockwise methods into our model. Since linear wavelet is not adaptive, nonlinear wavelet and design-adapted wavelet have also received considerable attention recently; see, for example, Li [50], Liang and Uña-Álvarez [51] and Chesneau [52] for (conditional) density estimator; Li and Xiao [20] and Uña-Álvarez et al.[53] for nonparametric models; and Delouille and Sachs [54] for nonlinear autoregressive models, and so on. At present, our paper mainly concentrates on linear wavelet estimator in nonparametric regression model with repeated measurements under weakly dependent processes. Although it is easy to construct nonlinear wavelet estimator in our model, it is very difficult to establish asymptotic theory of nonlinear wavelet estimator and to prove it since the structure of our model is complex. It will be a challenging work. Interesting work for further research includes nonlinear wavelet and design-adapted wavelet estimations for our model.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The three authors contributed equally to this work. All authors read and approved the final manuscript.

Acknowledgements

The authors thank the associate editor and two anonymous referees, whose valuable comments greatly improved the paper. This work is partially supported by Key Natural Science Foundation of Higher Education Institutions of Anhui Province (KJ2012A270), NSFC (11171065, 11061002), Youth Foundation for Humanities and Social Sciences Project from Ministry of Education of China (11YJC790311), NSFJS (BK2011058), National Natural Science Foundation of Guangxi (2011GXNSFA018126) and Postdoctoral Research Program of Jiangsu Province of China (1202013C).

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