Bernstein-Type Inequality for Widely Dependent Sequence and Its Application to Nonparametric Regression Models

and Applied Analysis 3 In this paper, we will present the Bernstein-type inequality for WD random variables. By using the Bernstein-type inequality, we will further investigate the strong consistency for the estimator of nonparametric regression models based on WD errors. This work is organized as follows: the Bernstein-type inequality for WD random variables is provided in Section 2 and strong consistency for the estimator of nonparametric regression models based on WD errors is investigated in Section 3. Throughout the paper, C denotes a positive constant not depending on n, which may be different in various places. a n = O(b n ) represents a n ≤ Cb n for all n ≥ 1. Let ⌈x⌉ denote the integer part of x and I(A) be the indicator function of the set A. Denote x+ = xI(x ≥ 0) and x− = −xI(x < 0). Let {ε n , n ≥ 1} be a sequence of WD random variables. Denote S n = ∑ n i=1 ε i . In the sequel, we will use the following different assumptions in different situations: lim n→∞ g (n) e −an c = 0, (8) lim n→∞ g (n) e −dlogn = 0, (9) where a, c, and d are finite positive constants. 2. Bernstein-Type Inequality for WD Random Variables In this section, we will present the Bernstein-type inequality for WD random variables, which will be used to prove the strong consistency for estimator of the nonparametric regression model based on WD random variables. Theorem 4. Let {ε n , n ≥ 1} be a sequence of WD random variables with Eε i = 0 and |ε i | ≤ b for each i ≥ 1, where b is a positive constant. Denote σ2 i = Eε 2 i and B2 n = ∑ n i=1 σ 2 i for each n ≥ 1. Then for any ε > 0, P (S n ≥ ε) ≤ g U (n) exp{− ε 2 2B n + (2/3) bε } , (10) P ( 󵄨󵄨󵄨Sn 󵄨󵄨󵄨 ≥ ε) ≤ 2g (n) exp{− ε 2 2B n + (2/3) bε } . (11) Proof. For any t > 0, by Taylor’s expansion, EX i = 0 and the inequality 1 + x ≤ ex for x ∈ R, we can get that for i = 1, 2, . . . , n, E exp {tε i } = 1 + ∞


Introduction
Let {  ,  ≥ 1} be a sequence of random variables defined on a fixed probability space (Ω, F, ).It is well known that the Bernstein-type inequality for the partial sum ∑  =1   plays an important role in probability limit theory and mathematical statistics.The main purpose of the paper is to present the Bernstein-type inequality, by which, we will further investigate the strong consistency for the estimator of nonparametric regression models based on widely dependent random variables.
The above estimator was first proposed by Georgiev [1] and subsequently has been studied by many authors.For instance, when   are assumed to be independent, consistency and asymptotic normality have been studied by Georgiev and Greblicki [2], Georgiev [3] and Müller [4] among others.Results for the case when   are dependent have also been studied by various authors in recent years.Fan [5] extended the work of Georgiev [3] and Müller [4] in the estimation of the regression model to the case where it forms an   -mixingale sequence for some 1 ≤  ≤ 2. Roussas [6] discussed strong consistency and quadratic mean consistency for   () under mixing conditions.Roussas et al. [7] established asymptotic normality of   () assuming that the errors are from a strictly stationary stochastic process and satisfying the strong mixing condition.Tran et al. [8] discussed again asymptotic normality of   () assuming that the errors form a linear time series, more precisely, a weakly stationary linear process based on a martingale difference sequence.Hu et al. [9] studied the asymptotic normality for double array sum of linear time series.Hu et al. [10] gave the mean consistency, complete consistency, and asymptotic normality of regression models with linear process errors.Liang and Jing [11] presented some asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences.Yang et al. [12] generalized the results of Liang and Jing [11] for negatively associated sequences to the case of negatively orthant dependent sequences and obtained the strong consistency for the estimator of the nonparametric regression models based on negatively orthant dependent errors.Wang et al. [13] studied the complete consistency of the estimator of nonparametric regression models based on ρ-mixing sequences, and so forth.The main purpose of this paper is to investigate the strong consistency for the estimator of the nonparametric regression models based on widely dependent random variables, which contains independent random variables, negatively associated random variables, negatively orthant dependent random variables, extended negatively orthant dependent random variables, and some positively dependent random variables as specials cases.For more details about the strong consistency for the estimator of (⋅), Ren and Chen [14] obtained the strong consistency for the least squares estimator of  and the nonparametric estimator of (⋅) based on negatively associated samples, Baek and Liang [15] studied the strong consistency for the weighted least squares estimator of  and nonparametric estimator of (⋅) in a semi-parametric model under negatively associated samples, which extended the corresponding one on independent random error settings, Liang et al. [16] also studied the strong consistency in a in semiparametric model for a linear process with negatively associated innovations and established the convergence rate, they also pointed out that their results on nonparametric estimator of (⋅) can attain the optimal convergence rate, and so forth.

Concepts of Wide Dependence.
In this section, we will present some wide dependence structures introduced in Wang et al. [17].
For examples of WD random variables with various dominating coefficients, we refer the reader to Wang et al. [17].These examples show that WD random variables contain some common negatively dependent random variables, some positively dependent random variables, and some others.For details about WD random variables, one can refer to Wang et al. [17], Wang and Cheng [18], Wang et al. [19], Chen et al. [20], and so forth.
Wang et al. [17] obtained the following properties for WD random variables, which will be used to prove the main results of the paper.
(2) If {  ,  ≥ 1} are WD, then for each  ≥ 1 and any  ∈ R, In this paper, we will present the Bernstein-type inequality for WD random variables.By using the Bernstein-type inequality, we will further investigate the strong consistency for the estimator of nonparametric regression models based on WD errors.
This work is organized as follows: the Bernstein-type inequality for WD random variables is provided in Section 2 and strong consistency for the estimator of nonparametric regression models based on WD errors is investigated in Section 3.
Throughout the paper,  denotes a positive constant not depending on , which may be different in various places.  = (  ) represents   ≤   for all  ≥ 1.Let ⌈⌉ denote the integer part of  and () be the indicator function of the set . Denote  + = ( ≥ 0) and  − = −( < 0).Let {  ,  ≥ 1} be a sequence of WD random variables.Denote   = ∑  =1   .In the sequel, we will use the following different assumptions in different situations: lim where , , and  are finite positive constants.

Bernstein-Type Inequality for WD Random Variables
In this section, we will present the Bernstein-type inequality for WD random variables, which will be used to prove the strong consistency for estimator of the nonparametric regression model based on WD random variables.
By Theorem 4, we can get the following complete convergence for WD random variables immediately.( Proof.For any  > 0, it follows from (11) that which implies (19).Here  and  1 are positive constants not depending on .

The Strong Consistency for the Estimator of Nonparametric Regression Models Based on WD Errors
Unless otherwise specified, we assume throughout the paper that   () is defined by (2).For any function (), we use () to denote all continuity points of the function  on .
The norm ‖‖ is the Euclidean norm.For any fixed design point  ∈ , the following assumptions on weight functions   () will be used: Since   =   = 0 for each , it is easy to see that By the condition ( 2 ), we can see that max for  large enough, (28) which implies by Borel-Cantelli lemma.
Next, we will estimate  2 and  3 .It can be checked by sup which implies Consequently, by Kronecker's lemma, we have that Combining ( 33) and (34), it follows that Likewise, by sup ≥1  2  < ∞, we can see that Hence, by Kronecker's lemma, From the statements above, we have Therefore, (24) follows from ( 26), ( 29), (35), and (41) immediately.This completes the proof of the theorem.
which implies by Borel-Cantelli lemma that Meanwhile, it can be checked by sup  (59) Based on the notations above, we can get the following result by using Theorems 6 and 7. (60) Hence, conditions ( 1 )-( 3 ) are satisfied.By Theorems 6 and 7, we can get (i) and (ii) immediately.This completes the proof of the corollary.