The Strong Consistency of the Estimator of Fixed-Design Regression Model under Negatively Dependent Sequences

and Applied Analysis 3 The norm ‖x‖ is the Eucledean norm. For any fixed design point x ∈ A, the following assumptions on weight function W ni (x) will be used: (A 1 ) ∑n i=1 W ni (x) → 1 as n → ∞; (A 2 ) ∑n i=1 |W ni (x)| ≤ C < ∞ for all n; (A 3 ) ∑n i=1 |W ni (x)| ⋅ |g(x ni ) − g(x)|I(‖x ni − x‖ > a) → 0 as n → ∞ for all a > 0. Based on the assumptions above, we can get the following strong consistency of the fixed design regression estimator g n (x). Theorem 6. Let {ε n , n ≥ 1} be a sequence of mean zero ND random variables, which is stochastically dominated by a random variable X. Assume that conditions (A 1 )–(A 3 ) hold true. If E|X|p < ∞ for some p > 1, and if there exist some s ∈ (1/p, 1) such that max 1≤i≤n 󵄨󵄨󵄨Wni (x) 󵄨󵄨󵄨 = O (n −s ) , (8) then for any x ∈ c(g), g n (x) 󳨀→ g (x) a.s., as n 󳨀→ ∞. (9) Proof. For x ∈ c(g) and a > 0, we have by (1) and (2) that 󵄨󵄨󵄨Egn (x) − g (x) 󵄨󵄨󵄨


Introduction
Let {  ,  ≥ 1} be a sequence of random variables defined on a fixed probability space (Ω, F, ).It is well known that the Rosenthal-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 investigate the strong consistency of the estimator of fixed design regression model under negatively dependent sequences, by using the Rosenthal-type inequality.
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 which form 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.Shen [13] presented the Bernstein-type inequality for widely dependent random variables and gave its application to nonparametric regression models, and so forth.The main purpose of this section is to investigate the strong consistency of estimator of the fixed design regression model based on negatively dependent random variables.
The concept of negatively dependent random variables was introduced by Lehmann [14] as follows.
A finite collection of random variables  1 ,  2 , . . .,   is said to be negatively dependent (or negatively orthant dependent, ND in short) if hold for all  1 ,  2 , . . .,   ∈ R.An infinite sequence {  ,  ≥ 1} is said to be ND if every finite subcollection is ND.
Obviously, independent random variables are ND.Joag-Dev and Proschan [15] pointed out that negatively associated (NA, in short) random variables are ND.They also presented an example in which  = ( 1 ,  2 ,  3 ,  4 ) possesses ND but does not possess NA.The another example which is ND but is not NA was provided by Wu [16] as follows.
So we can see that ND is weaker than NA.A number of well-known multivariate distributions have the ND properties, such as multinomial, convolution of unlike multinomials, multivariate hypergeometric, dirichlet, dirichlet compound multinomial, and multinomials having certain covariance matrices.Because of the wide applications of ND random variables, the limiting behaviors of ND random variables have received more and more attention recently.A number of useful results for ND random variables have been established by many authors.We refer to Volodin [17] for the Kolmogorov exponential inequality, Asadian et al. [18] for Rosenthal's type inequality, Kim [19] for Hájek-Rényi type inequality, Amini et al. [20,21], Ko and Kim [22], Klesov et al. [23], and Wang et al. [24] for almost sure convergence, Amini and Bozorgnia [25], Kuczmaszewska [26], Taylor et al. [27], Zarei and Jabbari [28], Wu [16,29], Sung [30], and Wang et al. [31] for complete convergence, Wang et al. [32] for exponential inequalities and inverse moment, Shen [33] for strong limit theorems for arrays of rowwise ND random variables, Shen [34] for strong convergence rate for weighted sums of arrays of rowwise ND random variables, and so on.When these are compared with the corresponding results of independent random variable sequences, there still remains much to be desired.This work is organized as follows: some preliminary lemmas are presented in Section 2, and the main results and their proofs are provided 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 let () be the indicator function of the set . Denote  + = ( ≥ 0) and  − = −( < 0).

Preliminaries
In this section, we will present some important lemmas which will be used to prove the main results of the paper.
Lemma 3 (cf.Asadian et al. [18]).Let  ≥ 2, and let {  ,  ≥ 1} be a sequence of ND random variables with   = 0 and |  |  < ∞ for every  ≥ 1.Then there exists a positive constant  depending only on  such that for every  ≥ 1, The following concept of stochastic domination will be used in this work.
By the definition of stochastic domination and integration by parts, we can get the following property for stochastic domination.For the details of the proof, one can refer to Wu [36,37] or Shen and Wu [38].
Lemma 5. Let {  ,  ≥ 1} be a sequence of random variables which is stochastically dominated by a random variable .For any  > 0 and  > 0, the following two statements hold: where  1 and  2 are positive constants.

Main Results and Their Proofs
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 Eucledean norm.For any fixed design point  ∈ , the following assumptions on weight function   () will be used: Based on the assumptions above, we can get the following strong consistency of the fixed design regression estimator   ().Theorem 6.Let {  ,  ≥ 1} be a sequence of mean zero ND random variables, which is stochastically dominated by a random variable .Assume that conditions ( 1 )-( 3 ) hold true.If ||  < ∞ for some  > 1, and if there exist some  ∈ (1/, 1) such that then for any  ∈ (),   () →  () a.s., as  → ∞.
Corollary 7. Let {  ,  ≥ 1} be a sequence of mean zero ND random variables, which is stochastically dominated by a random variable .Assume that  is continuous on the compact set .If ||  < ∞ for some  > 1 and if there exists some  ∈ (1/, 1) such that   = [  ], then for any  ∈ (), g () →  () a.s., as  → ∞. (32) Proof.It suffices to show that the conditions of Theorem 6 are satisfied.Since  is continuous on the compact set , hence  is uniformly continuous on the compact set , which implies that {|(  ) − ()| : 1 ≤  ≤ ,  ≥ 1} is bounded on the set .
For any  ∈ [0, 1], if follows from the definition of   () and W () that Hence, conditions (A 1 )-(A 3 ) and ( 8) are satisfied.By Theorem 6, we can get (32) immediately.This completes the proof of the corollary.

Definition 4 .
A sequence {  ,  ≥ 1} of random variables is said to be stochastically dominated by a random variable  if there exists a positive constant  such that  (           > ) ≤  (|| > )