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

Abstract

We study the strong consistency of estimator of fixed design regression model under negatively dependent sequences by using the classical Rosenthal-type inequality and the truncated method. As an application, the strong consistency for the nearest neighbor estimator is obtained.

1. Introduction

Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables defined on a fixed probability space (Ω,ℱ,𝑃). It is well known that the Rosenthal-type inequality for the partial sum ∑^𝑛_𝑖=1𝑋_𝑖 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.

Consider the following fixed design regression model: 𝑌_𝑛𝑖=𝑔𝑥_𝑛𝑖+𝜀_𝑛𝑖,𝑖=1,2,…,𝑛,𝑛≥1, (1) where 𝑥_𝑛𝑖 are known fixed design points from 𝐴, where 𝐴⊂ℝ^𝑝 is a given compact set for some 𝑝≥1, 𝑔(⋅) is an unknown regression function defined on 𝐴, and 𝜀_𝑛𝑖 are random errors. Assume that, for each 𝑛≥1, (𝜀_𝑛1,𝜀_𝑛2,…,𝜀_𝑛𝑛) have the same distribution as (𝜀₁,𝜀₂,…,𝜀_𝑛). As an estimator of 𝑔(⋅), the following weighted regression estimator will be considered: 𝑔_𝑛(𝑥)=^𝑛_𝑖=1𝑊_𝑛𝑖(𝑥)𝑌_𝑛𝑖,𝑥∈𝐴⊂ℝ^𝑝, (2) where 𝑊_𝑛𝑖(𝑥)=𝑊_𝑛𝑖(𝑥;𝑥_𝑛1,𝑥_𝑛2,…,𝑥_𝑛𝑛), 𝑖=1,2,…,𝑛, are the weight function.

The above estimator was first proposed by Georgiev [1] and subsequently has been studied by many authors. For instance, when 𝜀_𝑛𝑖 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 𝜀_𝑛𝑖 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 𝐿_𝑞-mixingale sequence for some 1≤𝑞≤2. Roussas [6] discussed strong consistency and quadratic mean consistency for 𝑔_𝑛(𝑥) under mixing conditions. Roussas et al. [7] established asymptotic normality of 𝑔_𝑛(𝑥) 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 𝑔_𝑛(𝑥) 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 𝑋₁,𝑋₂,…,𝑋_𝑛 is said to be negatively dependent (or negatively orthant dependent, ND in short) if 𝑃𝑋₁>𝑥₁,𝑋₂>𝑥₂,…,𝑋_𝑛>𝑥_𝑛≤^𝑛_𝑖=1𝑃𝑋_𝑖>𝑥_𝑖,𝑃𝑋₁≤𝑥₁,𝑋₂≤𝑥₂,…,𝑋_𝑛≤𝑥_𝑛≤^𝑛_𝑖=1𝑃𝑋_𝑖≤𝑥_𝑖 (3) hold for all 𝑥₁,𝑥₂,…,𝑥_𝑛∈ℝ. An infinite sequence {𝑋_𝑛,𝑛≥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 𝑋=(𝑋₁,𝑋₂,𝑋₃,𝑋₄) possesses ND but does not possess NA. The another example which is ND but is not NA was provided by Wu [16] as follows.

Example 1. Let 𝑋_𝑖 be a binary random variable such that 𝑃(𝑋_𝑖=1)=𝑃(𝑋_𝑖=0)=0.5 for 𝑖=1,2,3. Let (𝑋₁,𝑋₂,𝑋₃) take the values (0,0,1), (0,1,0), (1,0,0), and (1,1,1), each with probability 1/4. It can be verified that all the ND conditions hold. However, 𝑃𝑋₁+𝑋3≤1,𝑋₂=4≤08≰𝑃𝑋₁+𝑋3≤1𝑃𝑋₂=3≤08. (4) Hence, 𝑋₁,𝑋₂, and 𝑋₃ are not NA.

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, 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 let 𝐼(𝐴) be the indicator function of the set 𝐴. Denote 𝑥⁺=𝑥𝐼(𝑥≥0) and 𝑥⁻=−𝑥𝐼(𝑥<0).

2. Preliminaries

In this section, we will present some important lemmas which will be used to prove the main results of the paper.

Lemma 2 (cf. Bozorgnia et al. [35]). Let the random variables 𝑋₁,𝑋₂,…,𝑋_𝑛 be ND, and let 𝑓₁,𝑓₂,…,𝑓_𝑛 be all nondecreasing (or all nonincreasing) functions. Then random variables 𝑓₁(𝑋₁),𝑓₂(𝑋₂),…,𝑓_𝑛(𝑋_𝑛) are ND.

Lemma 3 (cf. Asadian et al. [18]). Let 𝑝≥2, and let {𝑋_𝑛,𝑛≥1} be a sequence of ND random variables with 𝐸𝑋_𝑛=0 and 𝐸|𝑋_𝑛|^𝑝<∞ for every 𝑛≥1. Then there exists a positive constant 𝐶 depending only on 𝑝 such that for every 𝑛≥1, 𝐸^𝑛_𝑖=1𝑋_𝑖^𝑝≤𝐶^𝑛_𝑖=1𝐸𝑋_𝑖^𝑝+^𝑛_𝑖=1𝐸𝑋²_𝑖^𝑝/2. (5)

The following concept of stochastic domination will be used in this work.

Definition 4. A sequence {𝑋_𝑛,𝑛≥1} of random variables is said to be stochastically dominated by a random variable 𝑋 if there exists a positive constant 𝐶 such that 𝑃𝑋_𝑛>𝑥≤𝐶𝑃(|𝑋|>𝑥) (6) for all 𝑥≥0 and 𝑛≥1.

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 {𝑋_𝑛,𝑛≥1} be a sequence of random variables which is stochastically dominated by a random variable 𝑋. For any 𝛼>0 and 𝑏>0, the following two statements hold: 𝐸𝑋_𝑛^𝛼𝐼𝑋_𝑛≤𝑏≤𝐶₁𝐸|𝑋|^𝛼𝐼(|𝑋|≤𝑏)+𝑏^𝛼𝑃𝑋|,𝐸𝑋(|>𝑏)_𝑛𝛼𝐼𝑋_𝑛>𝑏≤𝐶₂𝐸|𝑋|^𝛼𝐼(|𝑋|>𝑏), (7) where 𝐶₁ and 𝐶₂ are positive constants.

3. Main Results and Their Proofs

Unless otherwise specified, we assume throughout the paper that 𝑔_𝑛(𝑥) is defined by (2). For any function 𝑔(𝑥), we use 𝑐(𝑔) to denote all continuity points of the function 𝑔 on 𝐴. 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:(A₁)∑^𝑛_𝑖=1𝑊_𝑛𝑖(𝑥)→1 as 𝑛→∞;(A₂)∑^𝑛_𝑖=1|𝑊_𝑛𝑖(𝑥)|≤𝐶<∞ for all 𝑛;(A₃)∑^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 (𝐴₁)–(𝐴₃) 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}𝑊_𝑛𝑖(𝑥)=𝑂𝑛^−𝑠, (8) then for any 𝑥∈𝑐(𝑔), 𝑔_𝑛(𝑥)⟶𝑔(𝑥)𝑎.𝑠.,𝑎𝑠𝑛⟶∞. (9)

Proof. For 𝑥∈𝑐(𝑔) and 𝑎>0, we have by (1) and (2) that 𝐸𝑔_𝑛(𝑥)(𝑥≤−𝑔)_𝑛𝑖=1𝑊_𝑛𝑖(𝑥)⋅𝑔𝑥_𝑛𝑖−𝑔(𝑥)𝐼𝑥_𝑛𝑖+−𝑥≤𝑎_𝑛𝑖=1𝑊_𝑛𝑖(𝑥)⋅𝑔𝑥_𝑛𝑖−𝑔(𝑥)𝐼𝑥_𝑛𝑖+𝑔(𝑥⋅−𝑥>𝑎)^𝑛_𝑖=1𝑊_𝑛𝑖(𝑥)−1. (10) Since 𝑥∈𝑐(𝑔), hence for any 𝜀>0, there exist a 𝛿>0 and ‖𝑥^∗−𝑥‖<𝛿 such that |𝑔(𝑥^∗)−𝑔(𝑥)|<𝜀. If we take 𝑎∈(0,𝛿) in (10), we can get that 𝐸𝑔_𝑛(𝑥)−𝑔(𝑥)≤𝜀^𝑛_𝑖=1𝑊_𝑛𝑖(𝑥)+𝑔(𝑥)⋅^𝑛_𝑖=1𝑊_𝑛𝑖(𝑥)+−1_𝑛𝑖=1𝑊_𝑛𝑖(𝑥)⋅𝑔𝑥_𝑛𝑖−𝑔(𝑥)𝐼𝑥_𝑛𝑖−𝑥>𝑎. (11) Therefore, we have by conditions (A₁)–(A₃) that lim_𝑛→∞𝐸𝑔_𝑛(𝑥)=𝑔(𝑥),𝑥∈𝑐𝑔. (12) For a fixed design point 𝑥∈𝑐(𝑔), without loss of generality, we assume that 𝑊_𝑛𝑖(𝑥)≥0 (otherwise, we use 𝑊⁺_𝑛𝑖(𝑥) and 𝑊⁻_𝑛𝑖(𝑥) instead of 𝑊_𝑛𝑖(𝑥), resp., and note that 𝑊_𝑛𝑖(𝑥)=𝑊⁺_𝑛𝑖(𝑥)−𝑊⁻_𝑛𝑖(𝑥)).
By (12), we can see that in order to prove (9), we only need to show that 𝑔_𝑛(𝑥)−𝐸𝑔_𝑛(𝑥)=^𝑛_𝑖=1𝑊_𝑛𝑖(𝑥)𝜀_𝑛𝑖≐^𝑛_𝑖=1𝑅_𝑛𝑖⟶0a.s.,as𝑛⟶∞, (13) where 𝑅_𝑛𝑖=𝑊_𝑛𝑖(𝑥)𝜀_𝑛𝑖.
For fixed 𝜀>0, choose 1/𝑝<𝛿<𝑠 and some positive integer 𝑁 (to be specified later). Denote, for 𝑖=1,2,…,𝑛, 𝑋_𝑛𝑖(1)=−𝑛^−𝛿𝐼𝑅_𝑛𝑖<−𝑛^−𝛿+𝑅_𝑛𝑖𝐼𝑅_𝑛𝑖≤𝑛^−𝛿+𝑛^−𝛿𝐼𝑅_𝑛𝑖>𝑛^−𝛿,𝑋_𝑛𝑖(2)=𝑅_𝑛𝑖−𝑛^−𝛿𝐼𝑛^−𝛿<𝑅_𝑛𝑖<𝜀𝑁,𝑋_𝑛𝑖(3)=𝑅_𝑛𝑖+𝑛^−𝛿𝐼−𝑛^−𝛿>𝑅_𝑛𝑖𝜀>−𝑁,𝑋_𝑛𝑖(4)=𝑅_𝑛𝑖−𝑛^−𝛿𝐼𝑅_𝑛𝑖≥𝜀𝑁+𝑅_𝑛𝑖+𝑛^−𝛿𝐼𝑅_𝑛𝑖𝜀≤−𝑁. (14) It is easy to check that 𝑋_𝑛𝑖(1)+𝑋_𝑛𝑖(2)+𝑋_𝑛𝑖(3)+𝑋_𝑛𝑖(4)=𝑅_𝑛𝑖. Hence, to prove (13), it suffices to show that, for any 𝑥∈𝑐(𝑔), ^𝑛_𝑖=1𝑋_𝑛𝑖𝑗⟶0a.s.,as𝑛⟶∞,𝑗=1,2,3,4. (15)
By 𝐸𝜀_𝑛=0 and 𝐸|𝑋|^𝑝<∞, we can see that ^𝑛_𝑖=1𝐸𝑋_𝑛𝑖(1≤_𝑛𝑖=1)𝑛^−𝛿𝑃𝑅_𝑛𝑖>𝑛^−𝛿+𝐸𝑅_𝑛𝑖𝐼𝑅_𝑛𝑖>𝑛^−𝛿≤^𝑛_𝑖=1𝑛^−𝛿𝐸𝑅_𝑛𝑖^𝑝𝑛^−𝑝𝛿≤𝐶𝑛^{𝑛𝛿(𝑝−1)}_𝑖=1𝑊_𝑛𝑖(𝑥)^𝑝𝐸|𝑋|^𝑝≤𝐶𝑛^{𝛿(𝑝−1)}max_{1≤𝑖≤𝑛}𝑊_𝑛𝑖(𝑥)^𝑛𝑝−1_𝑖=1𝑊_𝑛𝑖(𝑥)≤𝐶𝑛^{−(𝑝−1)(𝑠−𝛿)}⟶0,as𝑛⟶∞. (16) Hence, to prove ∑^𝑛_𝑖=1𝑋_𝑛𝑖(1)→0a.s., it suffices to show that ^∞_𝑛=1𝑃^𝑛_𝑖=1𝑋_𝑛𝑖(1)−𝐸𝑋_𝑛𝑖(1)>𝜀<∞. (17) It is easily seen that, for fixed 𝑛≥1 and 𝑥∈𝑐(𝑔), {𝑋_𝑛𝑖(1),1≤𝑖≤𝑛} are still ND random variables by Lemma 2. Applying Lemma 3, we have for 𝑞≥2 that ^∞_𝑛=1𝑃^𝑛_𝑖=1𝑋_𝑛𝑖(1)−𝐸𝑋_𝑛𝑖(1)>𝜀≤𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑋_𝑛𝑖(1)^𝑞+𝐶^∞_𝑛=1^𝑛_𝑖=1𝐸𝑋_𝑛𝑖(1)^2𝑞/2≐𝐶𝐼₁₁+𝐶𝐼₁₂. (18) Taking 𝑞>max{𝑝,(𝑠+1+𝑝(𝛿−𝑠))/𝛿,2/((𝑠−𝛿)(𝑝−1)+𝛿),2/𝑠}, we can get that 𝑞−𝑝𝛿+𝑠𝑝−1>1,𝑞𝑠2𝑞>1,2−𝑝𝛿+𝑠𝑝−12=(𝑠−𝛿)𝑝−1+𝛿𝑞2>1. (19)
For 𝐼₁₁, we have by 𝐶_𝑟's inequality, Lemma 5, and condition (A₂) that 𝐼₁₁≤𝐶^∞𝑛_𝑛=1𝑖=1𝑛^−𝑞𝛿𝑃𝑅_𝑛𝑖>𝑛^−𝛿+𝐸𝑅_𝑛𝑖^𝑞𝐼𝑅_𝑛𝑖≤𝑛^−𝛿≤𝐶^∞𝑛_𝑛=1𝑖=1𝑛^−𝑞𝛿𝑃𝑊_𝑛𝑖(𝑥)𝑋>𝑛^−𝛿+𝐸𝑊_𝑛𝑖(𝑥)𝑋^𝑞𝐼𝑊_𝑛𝑖(𝑥)𝑋≤𝑛^−𝛿≤𝐶^∞𝑛_𝑛=1𝑖=1𝑛^−𝑞𝛿𝐸𝑊_𝑛𝑖(𝑥)𝑋^𝑝𝑛^−𝑝𝛿≤𝐶^∞_𝑛=1𝑛^{−(𝑞−𝑝)𝛿}max_{1≤𝑖≤𝑛}𝑊_𝑛𝑖(𝑥)^𝑝−1×^𝑛_𝑖=1𝑊_𝑛𝑖(𝑥)≤𝐶^∞_𝑛=1𝑛^{−[(𝑞−𝑝)𝛿+𝑠(𝑝−1)]}<∞. (20)
For 𝐼₁₂, if 1<𝑝<2, we have by the proof of (20) that 𝐼₁₂≤𝐶^∞_𝑛=1^𝑛_𝑖=1𝑛^−2𝛿𝑃𝑅_𝑛𝑖>𝑛^−𝛿+𝐸𝑅_𝑛𝑖²𝐼𝑅_𝑛𝑖≤𝑛^{−𝛿𝑞/2}≤𝐶^∞_𝑛=1𝑛^{−[(2−𝑝)𝛿+𝑠(𝑝−1)]𝑞/2}<∞; (21) if 𝑝≥2, we have by 𝐸𝑋²<∞ that 𝐼₁₂≤𝐶^∞_𝑛=1^𝑛_𝑖=1𝑛^−2𝛿𝑃𝑅_𝑛𝑖>𝑛^−𝛿+𝐸𝑅_𝑛𝑖²𝐼𝑅_𝑛𝑖≤𝑛^{−𝛿𝑞/2}≤𝐶^∞_𝑛=1^𝑛_𝑖=1𝑛^−2𝛿𝑃𝑊_𝑛𝑖(𝑥)𝑋>𝑛^−𝛿+𝑊²_𝑛𝑖(𝑥)^𝑞/2≤𝐶^∞_𝑛=1𝑛^{−[(2−𝑝)𝛿+𝑠(𝑝−1)]𝑞/2}+𝐶^∞_𝑛=1𝑛^{−𝑞𝑠/2}<∞. (22) By (18)–(22), we can get (17), which together with (16) imply that ∑^𝑛_𝑖=1𝑋_𝑛𝑖(1)→0a.s.
Next, we will prove that ∑^𝑛_𝑖=1𝑋_𝑛𝑖(2)→0a.s. Since 0≤𝑋_𝑛𝑖(2)<𝜀/𝑁, |∑^𝑛_𝑖=1𝑋_𝑛𝑖(2)|=∑^𝑛_𝑖=1𝑋_𝑛𝑖(2)>𝜀 implies that there are at least 𝑁  𝑖’s such that 𝑋_𝑛𝑖(2)≠0. Hence, 𝑃^𝑛_𝑖=1𝑋_𝑛𝑖(2)>𝜀≤𝑃thereareatleast𝑁𝑖'ssuchthat𝑋_𝑛𝑖(2)≠0≤_{1≤𝑖1<𝑖2<⋯<𝑖𝑁≤𝑛}𝑃𝑋_𝑛𝑖1(2)≠0,𝑋_𝑛𝑖2(2)≠0,…,𝑋_𝑛𝑖𝑁(2)≠0≤_{1≤𝑖1<𝑖2<⋯<𝑖𝑁≤𝑛}𝑃𝑅_𝑛𝑖1>𝑛^−𝛿,𝑅_𝑛𝑖2>𝑛^−𝛿,…,𝑅_𝑛𝑖𝑁>𝑛^−𝛿≤_{1≤𝑖1<𝑖2<⋯<𝑖𝑁≤𝑛}𝑃𝑅_𝑛𝑖1>𝑛^−𝛿𝑃𝑅_𝑛𝑖2>𝑛^−𝛿⋯𝑃𝑅_𝑛𝑖𝑁>𝑛^−𝛿≤^𝑛_𝑖=1𝑃𝑅_𝑛𝑖>𝑛^−𝛿𝑁≤^𝑛_𝑖=1𝑃𝑅_𝑛𝑖>𝑛^−𝛿𝑁≤𝐶^𝑛_𝑖=1𝐸𝑊_𝑛𝑖(𝑥)𝑋^𝑝𝑛^{−𝑝𝛿𝑁}≤𝐶𝑛^𝑝𝛿max_{1≤𝑖≤𝑛}𝑊_𝑛𝑖(𝑥)^𝑛𝑝−1_𝑖=1𝑊_𝑛𝑖(𝑥)^𝑁≤𝐶𝑛^{−[(𝑠−𝛿)𝑝+𝑠]𝑁}, (23) which is summable if we choose large integer 𝑁 such that [(𝑠−𝛿)𝑝+𝑠]𝑁>1. Hence, ∑^𝑛_𝑖=1𝑋_𝑛𝑖(2)→0a.s. by Borel-Cantelli lemma.
Since −𝜀/𝑁<𝑋_𝑛𝑖(3)≤0, |∑^𝑛_𝑖=1𝑋_𝑛𝑖(3)|=−∑^𝑛_𝑖=1𝑋_𝑛𝑖(3)>𝜀 implies that there are at least 𝑁𝑖’s such that 𝑋_𝑛𝑖(3)≠0. Similarly, we have ∑^𝑛_𝑖=1𝑋_𝑛𝑖(3)→0a.s. as 𝑛→∞.
Finally, we will prove that ∑^𝑛_𝑖=1𝑋_𝑛𝑖(4)→0a.s. as 𝑛→∞. It is easily seen that ^𝑛_𝑖=1𝑋_𝑛𝑖(4≤_𝑛𝑖=1)𝑅_𝑛𝑖𝐼𝑅_𝑛𝑖≥𝜀𝑁+𝑛^𝑛−𝛿_𝑖=1𝐼𝑅_𝑛𝑖≥𝜀𝑁≤𝐶𝑛^𝑛−𝑠_𝑖=1𝜀_𝑛𝑖𝐼𝜀_𝑛𝑖≥𝐶𝑛^𝑠+𝑛^𝑛−𝛿_𝑖=1𝐼𝜀_𝑛𝑖≥𝐶𝑛^𝑠≤𝐶𝑛^𝑛−𝑠_𝑖=1𝜀_𝑛𝑖𝐼𝜀_𝑛𝑖≥𝐶𝑖^𝑠+𝑛^𝑛−𝛿_𝑖=1𝐼𝜀_𝑛𝑖≥𝐶𝑖^𝑠≐𝐻_𝑛1+𝐻_𝑛2. (24) To prove that ∑^𝑛_𝑖=1𝑋_𝑛𝑖(4)→0a.s. as 𝑛→∞, it suffices to show that 𝐻_𝑛1→0a.s. and 𝐻_𝑛2→0a.s. as 𝑛→∞. Firstly, we will show that ^∞_𝑖=1𝑖^−𝑠𝜀_𝑛𝑖𝐼𝜀_𝑛𝑖≥𝐶𝑖^𝑠<∞a.s.,_∞𝑖=1𝑖^−𝛿𝐼𝜀_𝑛𝑖≥𝐶𝑖^𝑠<∞a.s. (25)(26) Denote that 𝑇_𝑚=∑^𝑚_𝑖=1𝑖^−𝑠|𝜀_𝑛𝑖|𝐼(|𝜀_𝑛𝑖|≥𝐶𝑖^𝑠). For 𝑚≥𝑘≥1, it follows by Lemma 5 and 𝐸|𝑋|^𝑝<∞ that 𝐸𝑇_𝑚−𝑇_𝑘=^𝑚_𝑖=𝑘+1𝑖^−𝑠𝐸𝜀_𝑛𝑖𝐼𝜀_𝑛𝑖≥𝐶𝑖^𝑠≤𝐶^𝑚_𝑖=𝑘+1𝑖^−𝑠𝐸|𝑋|𝐼|𝑋|≥𝐶𝑖^𝑠≤𝐶^𝑚_𝑖=𝑘+1𝑖^−𝑠𝑝≤𝐶𝑘^{−𝑠𝑝+1}⟶0as𝑘⟶∞, (27) which implies that {𝑇_𝑚,𝑚≥1} is a Cauchy sequence in 𝐿₁, and hence there exists a random variable 𝑇 such that 𝐸|𝑇|<∞ and 𝐸|𝑇_𝑚−𝑇|→0. It follows by Lemma 5 and 𝐸|𝑋|^𝑝<∞ again that 𝑃𝑇₂^𝑘𝑇−𝑇>𝜀≤𝐶𝐸₂^𝑘−𝑇≤𝐶limsup_𝑚→∞𝐸𝑇₂^𝑘−𝑇_𝑚≤𝐶^∞_𝑖=2𝑘+1𝑖^−𝑠𝐸𝜀_𝑛𝑖𝐼𝜀_𝑛𝑖≥𝐶𝑖^𝑠≤𝐶^∞_𝑖=2𝑘+1𝑖^−𝑠𝐸|𝑋|𝐼|𝑋|≥𝐶𝑖^𝑠≤𝐶^∞_𝑖=2𝑘+1𝑖^−𝑠𝑝≤𝐶2^{−𝑘(𝑠𝑝−1)},𝑃max_{2𝑘−1<𝑚≤2𝑘}𝑇_𝑚−𝑇₂^𝑘−1>𝜀≤𝐶𝐸max_{2𝑘−1<𝑚≤2𝑘}𝑇_𝑚−𝑇₂^𝑘−1≤𝐶^2𝑘_{𝑖=2𝑘−1+1}𝑖^−𝑠𝐸𝜀_𝑛𝑖𝐼𝜀_𝑛𝑖≥𝐶𝑖^𝑠≤𝐶^2𝑘_{𝑖=2𝑘−1+1}𝑖^−𝑠𝑝≤𝐶2^{−𝑘(𝑠𝑝−1)}, (28) which together imply that 𝑇_𝑚→𝑇a.s., and thus (25) holds. Similarly, we can get (26). Therefore, 𝐻_𝑛1→0a.s. and 𝐻_𝑛2→0a.s. follow by (25)-(26) and the Kronecker's lemma immediately. This completes the proof of the theorem.

As an application of Theorem 6, we give the strong consistency for the nearest neighbor estimator of 𝑔(𝑥). Without loss of generality, putting 𝐴=[0,1], taking 𝑥_𝑛𝑖=𝑖/𝑛, 𝑖=1,2,…,𝑛, and 𝑛≥1. For any 𝑥∈𝐴, we rewrite |𝑥_𝑛1−𝑥|,|𝑥_𝑛2−𝑥|,…,|𝑥_𝑛𝑛−𝑥| as follows: 𝑥_𝑅^(𝑛)_1(𝑥)−𝑥≤𝑥_𝑅^(𝑛)_2(𝑥)−𝑥≤⋯≤𝑥_𝑅^(𝑛)_𝑛(𝑥)−𝑥. (29) If |𝑥_𝑛𝑖−𝑥|=|𝑥_𝑛𝑗−𝑥|, then |𝑥_𝑛𝑖−𝑥| is permuted before |𝑥_𝑛𝑗−𝑥| when 𝑥_𝑛𝑖<𝑥_𝑛𝑗.

Let 1≤𝑘_𝑛≤𝑛, the nearest neighbor weight function estimator of 𝑔(𝑥) in model (1) is defined as follows: 𝑔_𝑛(𝑥)=^𝑛_𝑖=1𝑊_𝑛𝑖(𝑥)𝑌_𝑛𝑖, (30) where 𝑊_𝑛𝑖(𝑥)=1𝑘_𝑛,if𝑥_𝑛𝑖−𝑥≤𝑥_𝑅^(𝑛)_{𝑘𝑛(𝑥)}−𝑥,0,otherwise. (31)

Based on the notations above, we can get the following result by using Theorem 6.

Corollary 7. 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 𝑔 is continuous on the compact set 𝐴. If 𝐸|𝑋|^𝑝<∞ for some 𝑝>1 and if there exists some 𝑠∈(1/𝑝,1) such that 𝑘_𝑛=[𝑛^𝑠], then for any 𝑥∈𝑐(𝑔), 𝑔_𝑛(𝑥)⟶𝑔(𝑥)𝑎.𝑠.,𝑎𝑠𝑛⟶∞. (32)

Proof. It suffices to show that the conditions of Theorem 6 are satisfied. Since 𝑔 is continuous on the compact set 𝐴, hence 𝑔 is uniformly continuous on the compact set 𝐴, which implies that {|𝑔(𝑥_𝑛𝑖)−𝑔(𝑥)|∶1≤𝑖≤𝑛,𝑛≥1} is bounded on the set 𝐴.
For any 𝑥∈[0,1], if follows from the definition of 𝑅_𝑖(𝑥) and 𝑊_𝑛𝑖(𝑥) that ^𝑛_𝑖=1𝑊_𝑛𝑖(𝑥)=^𝑛_𝑖=1𝑊_{𝑛𝑅𝑖(𝑥)}(𝑥)=^𝑘𝑛_𝑖=11𝑘_𝑛=1,max_{1≤𝑖≤𝑛}𝑊_𝑛𝑖(𝑥)=1𝑘_𝑛,𝑊_𝑛𝑖(𝑥)≥0,_𝑛𝑖=1𝑊_𝑛𝑖(𝑥)⋅𝑔𝑥_𝑛𝑖−𝑔(𝑥)𝐼𝑥_𝑛𝑖−𝑥>𝑎≤𝐶^𝑛_𝑖=1𝑥_𝑛𝑖−𝑥²𝑊_𝑛𝑖(𝑥)𝑎²=𝐶^𝑘𝑛_𝑖=1𝑥_𝑅^(𝑛)_𝑖(𝑥)−𝑥²𝑘_𝑛𝑎²≤𝐶^𝑘𝑛_𝑖=1(𝑖/𝑛)²𝑘_𝑛𝑎²𝑘≤𝐶^𝑛𝑛𝑎²,∀𝑎>0. (33) Hence, conditions (A₁)–(A₃) and (8) are satisfied. By Theorem 6, we can get (32) immediately. This completes the proof of the corollary.

Acknowledgments

The authors are most grateful to the Editor Michiel Bertsch and anonymous referee for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the National Natural Science Foundation of China (11201001, 11171001), the Natural Science Foundation of Anhui Province (1208085QA03, 1308085QA03), Applied Teaching Model Curriculum of Anhui University (XJYYXKC04), Doctoral Research Start-up Funds Projects of Anhui University, Students Innovative Training Project of Anhui University (201310357004), and the Students Science Research Training Program of Anhui University (KYXL2012007, kyxl2013003).