On consistency of the weighted least squares estimators in a semiparametric regression model

Abstract

This paper is concerned with the semiparametric regression model $y_i=x_i\beta +g(t_i)+\sigma _ie_i,~~i=1,2,\ldots ,n,$ where $\sigma _i^2=f(u_i)$, $(x_i,t_i,u_i)$ are known fixed design points, $\beta$ is an unknown parameter to be estimated, $g(\cdot )$ and $f(\cdot )$ are unknown functions, random errors $e_i$ are widely orthant dependent random variables. The p-th ($p>0$) mean consistency and strong consistency for least squares estimators and weighted least squares estimators of $\beta$ and g under some more mild conditions are investigated. A simulation study is also undertaken to assess the finite sample performance of the results that we established. The results obtained in the paper generalize and improve some corresponding ones of negatively associated random variables.

Appendix

Lemma A.1

Let $p>0$ and $\{X_n,n\ge 1\}$ be a sequence of zero mean WOD random variables with dominating coefficient h(n), which is stochastically dominated by a random variable X. Assume that $\{a_{ni}(\cdot ), 1\le i\le n, n\ge 1\}$ is a function array defined on compact set A satisfying

$$\begin{aligned} \max _{1\le j\le n}\sum _{i=1}^n |a_{ni}(z_j)|=O(1) \end{aligned}$$

(3.15)

and

$$\begin{aligned} \max _{1\le i,j \le n}|a_{ni}(z_j)|=O(n^{-\alpha }(h(n))^{-\beta }),~\exists ~\alpha >0,~\beta \ge 1. \end{aligned}$$

(3.16)

If $EX^2<\infty$ for $0<p\le 2$, then

$$\begin{aligned} \lim _{n\rightarrow \infty }\max _{1\le j\le n}E\left| \sum _{i=1}^na_{ni}(z_j)X_i\right| ^p=0. \end{aligned}$$

(3.17)

If $E|X|^p<\infty$ for $p>2$, then (3.17) still holds.

Remark A.1

Lemma A.1 also holds when the moment condition $EX^2<\infty$ is changed to $\sup _{i}EX_i^2<\infty$, $E|X|^p<\infty$ is changed to $\sup _{i}E|X_i|^p<\infty$ and the condition of stochastic domination is deleted. Under the similar modification, Theorem 2.1 also holds true.

Proof of Lemma A.1

Without loss of generality, we can assume that $a_{ni}(z_j)>0$.

If $0<p\le 2$, by Jensen’s inequality, Marcinkiewicz-Zygmund-type inequality (one can refer to Wang et al. (2014) for instance), (3.15), (3.16) and $EX^2<\infty$, we have

$$\begin{aligned}&\max _{1\le j\le n}E\left| \sum _{i=1}^na_{ni}(z_j)X_i\right| ^p \\&\quad \le C\left( EX^2\right) ^{p/2}\left( h(n)\max _{1\le i,j\le n}a_{ni}(z_j)\right) ^{p/2}\left( \max _{1\le j\le n}\sum _{i=1}^na_{ni}(z_j)\right) ^{p/2}\\&\quad \le Cn^{-\alpha p/2}(h(n))^{(1-\beta )p/2}\rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

If $p>2$, we denote

$$\begin{aligned} X_{ni}^{j}=n^{1/p}(h(n))^{\beta /p}a_{ni}(z_j)X_i, \end{aligned}$$

thus, we only need to prove

$$\begin{aligned} \frac{1}{n(h(n))^{\beta }}\max _{1\le j\le n}E\left| \sum _{i=1}^nX_{ni}^j\right| ^p\rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

For any $t>0$, denote

$$\begin{aligned}&Y_{ni}^j=-t^{1/p}I(X_{ni}^j<-t^{1/p})+X_{ni}^jI(|X_{ni}^j|\le t^{1/p})+t^{1/p}I(X_{ni}^j>t^{1/p}),\\&Z_{ni}^j=(X_{ni}^j+t^{1/p})I(X_{ni}^j<-t^{1/p}) +(X_{ni}^j-t^{1/p})I(X_{ni}^j>t^{1/p}). \end{aligned}$$

For fixed $t>0$ and $1 \le j\le n$, we can see that $\{Y_{ni}^j,1\le i\le n, n\ge 1\}$ and $\{Z_{ni}^j,1\le i\le n, n\ge 1\}$ are both arrays of rowwise WOD random variables. Noting that $X_{ni}^j=Y_{ni}^j-EY_{ni}^j+Z_{ni}^j-EZ_{ni}^j$, we have

$$\begin{aligned}&\frac{1}{n(h(n))^{\beta }}\max _{1\le j\le n}E\left| \sum _{i=1}^nX_{ni}^j\right| ^p\nonumber \\&\quad =\frac{1}{n(h(n))^{\beta }}\max _{1\le j\le n}\left[ \int _{0}^{n\varepsilon }P\left( \left| \sum _{i=1}^nX_{ni}^j\right| ^p>t\right) dt\right. \nonumber \\&\left. \qquad +\int _{n\varepsilon }^{\infty }P\left( \left| \sum _{i=1}^nX_{ni}^j\right| ^p>t\right) dt\right] \nonumber \\&\quad \le \varepsilon +\frac{1}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }P\left( \left| \sum _{i=1}^n\left( Y_{ni}^j-E Y_{ni}^j\right) \right|>t^{1/p}/2\right) dt\nonumber \\&\qquad +\frac{1}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }P\left( \left| \sum _{i=1}^n\left( Z_{ni}^j-E Z_{ni}^j\right) \right| >t^{1/p}/2\right) dt\nonumber \\&\quad \doteq \varepsilon +I_1+I_2. \end{aligned}$$

(3.18)

First, we prove $I_2\rightarrow 0,~n\rightarrow \infty$. Note that

$$\begin{aligned} \max _{1\le j\le n}\max _{t>n\varepsilon }\left| t^{-1/p}\sum _{i=1}^nEZ_{ni}^j\right|\le & {} C n^{-1} \max _{1\le j\le n}\sum _{i=1}^nE|X_{ni}^j|^pI\left( |X_{ni}^j|>(n\varepsilon )^{1/p}\right) \\\le & {} C (h(n))^{\beta }\max _{1\le j\le n}\sum _{i=1}^na_{ni}^p(z_j)E|X_i|^p\\\le & {} CE|X|^p(h(n))^{\beta }\max _{1\le i,j\le n}a_{ni}^{p-1}(z_j)\max _{1\le j\le n}\sum _{i=1}^na_{ni}(z_j)\\\le & {} Cn^{-\alpha (p-1)}(h(n))^{-\beta (p-2)}E|X|^p\rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

Hence, for any $t>n\varepsilon$ and all n large enough, we have $\max _{1\le j\le n}\left| \sum \nolimits _{i=1}^nEZ_{ni}^j\right| \le t^{1/p}/4$, which implies that for all n large enough,

$$\begin{aligned} I_2\le & {} \frac{1}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }P\left( \left| \sum _{i=1}^nZ_{ni}^j\right|>t^{1/p}/4\right) dt\nonumber \\\le & {} \frac{1}{n(h(n))^{\beta }}\max _{1\le j\le n}\sum _{i=1}^n\int _{n\varepsilon }^{\infty }P\left( |X_{ni}^j|>t^{1/p}\right) dt\nonumber \\\le & {} \frac{1}{n(h(n))^{\beta }}\max _{1\le j\le n}\sum _{i=1}^nE|X_{ni}^j|^pI(|X_{ni}^j|^p>n\varepsilon )\nonumber \\\le & {} \max _{1\le j\le n}\sum _{i=1}^na_{ni}^p(z_j)E|X_i|^p\le CE|X|^p\max _{1\le i,j\le n}a_{ni}^{p-1}(z_j)\max _{1\le j\le n}\sum _{i=1}^na_{ni}(z_j)\nonumber \\\le & {} CE|X|^pn^{-\alpha (p-1)}(h(n))^{-\beta (p-1)}\rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

(3.19)

Next, we will show that $I_1\rightarrow 0,~n\rightarrow \infty$. Taking $q>p$, we have by Markov’s inequality and Rosenthal-type inequality (one can refer to Wang et al. (2014) for instance) that

$$\begin{aligned} I_1\le & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }t^{-q/p}E\left| \sum _{i=1}^n\left( Y_{ni}^j-E Y_{ni}^j\right) \right| ^qdt\nonumber \\\le & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }t^{-q/p}\sum _{i=1}^nE|Y_{ni}^j|^qdt\nonumber \\&\quad +\frac{C h(n)}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }t^{-q/p}\left( \sum _{i=1}^nE\left( Y_{ni}^j\right) ^2\right) ^{q/2}dt\nonumber \\\doteq & {} I_{11}+I_{12}. \end{aligned}$$

(3.20)

According to the definition of $Y_{ni}^j$, we have

$$\begin{aligned} I_{11}\le & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }t^{-q/p}\sum _{i=1}^nE|X_{ni}^j|^qI\left( |X_{ni}^j|\le t^{1/p}\right) dt\nonumber \\&+\,\frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }\sum _{i=1}^nP\left( |X_{ni}^j|>t^{1/p}\right) dt\nonumber \\\doteq & {} I_{111}+I_{112}. \end{aligned}$$

(3.21)

In view of the proof of $I_2$, we can get that $I_{112}\rightarrow 0,~n\rightarrow \infty$. Next, we estimate the limit of $I_{111}$ as $n\rightarrow \infty$. It is easy to check that

$$\begin{aligned} I_{111}\le & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }t^{-q/p}\sum _{i=1}^nE|X_{ni}^j|^qI\left( |X_{ni}^j|^p\le (n+1)\varepsilon \right) dt\nonumber \\&+\,\frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }t^{-q/p}\sum _{i=1}^nE|X_{ni}^j|^qI\left( (n+1)\varepsilon<|X_{ni}^j|^p\le t\right) dt\nonumber \\= & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }t^{-q/p}\sum _{i=1}^nE|X_{ni}^j|^qI\left( |X_{ni}^j|^p\le (n+1)\varepsilon \right) dt\nonumber \\&+\,\frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{(n+1)\varepsilon }^{\infty }t^{-q/p}\sum _{i=1}^nE|X_{ni}^j|^qI\left( (n+1)\varepsilon <|X_{ni}^j|^p\le t\right) dt\nonumber \\\doteq & {} I_{111}^{'}+I_{111}^{''}. \end{aligned}$$

(3.22)

Similar to the proof of (3.19), we have

$$\begin{aligned} I_{111}^{'}\le & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\sum _{i=1}^nE|X_{ni}^j|^pI\left( |X_{ni}^j|^p\le (n+1)\varepsilon \right) \nonumber \\\le & {} C\max _{1\le j\le n}\sum _{i=1}^na_{ni}^p(z_j)E|X_i|^p\rightarrow 0,~~n\rightarrow \infty , \end{aligned}$$

(3.23)

and

$$\begin{aligned} I_{111}^{''}= & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\sum _{m=n+1}^{\infty }\int _{m\varepsilon }^{(m+1)\varepsilon }t^{-q/p}\sum _{i=1}^nE|X_{ni}^j|^qI((n+1)\varepsilon<|X_{ni}^j|^p\nonumber \\\le & {} t)dt\nonumber \\\le & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\sum _{m=n+1}^{\infty }m^{-q/p}\sum _{i=1}^nE|X_{ni}^j|^qI((n+1)\varepsilon<|X_{ni}^j|^p\le (m+1)\varepsilon )\nonumber \\= & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\sum _{m=n+1}^{\infty }m^{-q/p}\sum _{i=1}^n\sum _{k=n+1}^{m}E|X_{ni}^j|^qI(k\varepsilon<|X_{ni}^j|^p\le (k+1)\varepsilon )\nonumber \\\le & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\sum _{i=1}^n\sum _{k=n+1}^{\infty }k^{1-q/p}E|X_{ni}^j|^qI(k\varepsilon <|X_{ni}^j|^p\le (k+1)\varepsilon )\nonumber \\\le & {} \frac{C}{n(h(n))^{\beta }}\max _{1\le j\le n}\sum _{i=1}^nE|X_{ni}^j|^p\rightarrow 0,~~n\rightarrow \infty , \end{aligned}$$

(3.24)

which imply that $I_{111}\rightarrow 0,~n\rightarrow \infty$. Noting that $p>2$, $\beta \ge 1$ and $EX^2<\infty$, we have

$$\begin{aligned} I_{12}\le & {} \frac{C h(n)}{n(h(n))^{\beta }}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }t^{-q/p}\left( \sum _{i=1}^nE\left( X_{ni}^j\right) ^2I\left( |X_{ni}^j|\le t^{1/p}\right) \right. \nonumber \\&\left. +\sum _{i=1}^nt^{2/p}P\left( |X_{ni}^j|>t^{1/p}\right) \right) ^{q/2}dt\nonumber \\\le & {} \frac{C}{n}\max _{1\le j\le n}\int _{n\varepsilon }^{\infty }t^{-q/p}\left( \sum _{i=1}^nE\left( X_{ni}^j\right) ^2\right) ^{q/2}dt\nonumber \\\le & {} \frac{C}{n}n^{1-q/p}\max _{1\le j\le n}\left( \sum _{i=1}^nE\left( n^{1/p}(h(n))^{\beta /p}a_{ni}(z_j)X_i\right) ^2\right) ^{q/2}\nonumber \\\le & {} C(EX^2)^{q/2}n^{-\alpha q/2}(h(n))^{(1/p-1/2)\beta q}\rightarrow 0,~~n\rightarrow \infty . \end{aligned}$$

(3.25)

The proof is completed. $\square$

Lemma A.2

Let $\{X_n,n\ge 1\}$ be a sequence of zero mean WOD random variables with dominating coefficient h(n), which is stochastically dominated by a random variable X. Assume that $\{a_{ni}(\cdot ), 1\le i\le n, n\ge 1\}$ is a function array defined on compact set A satisfying

$$\begin{aligned} \max _{1\le j\le n}\sum _{i=1}^n |a_{ni}(z_j)|=O(1) \end{aligned}$$

(3.26)

and

$$\begin{aligned} \max _{1\le i,j\le n}|a_{ni}(z_j)|=O(n^{-\alpha }(h(n))^{-\beta }),~\exists ~\alpha>0,~\beta >0. \end{aligned}$$

(3.27)

If $EX^2<\infty$ and $\sum \nolimits _{i=1}^{n} i^{-\alpha }(h(i))^{-\beta }=O(n^{\alpha })$ for some $\alpha >0$ and $\beta >0$, then

$$\begin{aligned} \max _{1\le j\le n}\left| \sum _{i=1}^na_{ni}(z_j)X_i\right| \rightarrow 0~~a.s.,~n\rightarrow \infty . \end{aligned}$$

(3.28)

Proof

Without loss of generality, we can assume that $a_{ni}(z_j)>0$.

For any $\varepsilon >0$, choose $0<\delta <\alpha /2$ and large $N\ge 1$, which will be specialized later. Denote $X_{ni}(j)=a_{ni}(z_j)X_i$, and

$$\begin{aligned} Y_{ni}^{(1)}(j)= & {} -n^{-\delta }(h(n))^{-\beta /4}I\left( X_{ni}(j)<-n^{-\delta }(h(n))^{-\beta /4}\right) \\&+\,X_{ni}(j)I\left( |X_{ni}(j)|\le n^{-\delta }(h(n))^{-\beta /4}\right) \\&+\,n^{-\delta }(h(n))^{-\beta /4}I\left( X_{ni}(j)>n^{-\delta }(h(n))^{-\beta /4}\right) ,\\ Y_{ni}^{(2)}(j)= & {} \left( X_{ni}(j)+n^{-\delta }(h(n))^{-\beta /4}\right) I\left( X_{ni}(j)\le -\frac{\varepsilon }{N}(h(n))^{-\beta /4}\right) \\&+\,\left( X_{ni}(j)-n^{-\delta }(h(n))^{-\beta /4}\right) I\left( X_{ni}(j)\ge \frac{\varepsilon }{N}(h(n))^{-\beta /4}\right) ,\\ Y_{ni}^{(3)}(j)= & {} \left( X_{ni}(j)-n^{-\delta }(h(n))^{-\beta /4}\right) I\left( n^{-\delta }(h(n))^{-\beta /4}\le X_{ni}(j)<\frac{\varepsilon }{N}(h(n))^{-\beta /4}\right) ,\\ Y_{ni}^{(4)}(j)= & {} \left( X_{ni}(j)+n^{-\delta }(h(n))^{-\beta /4}\right) I\left( -\frac{\varepsilon }{N}(h(n))^{-\beta /4}\right) <X_{ni}(j)\\\le & {} -n^{-\delta }\left( h(n))^{-\beta /4}\right) . \end{aligned}$$

Then

$$\begin{aligned} \max _{1\le j\le n}\left| \sum _{i=1}^n a_{ni}(z_j)X_i\right|\le & {} \max _{1\le j\le n}\left| \sum _{i=1}^n Y_{ni}^{(1)}(j)\right| +\max _{1\le j\le n}\left| \sum _{i=1}^n Y_{ni}^{(2)}(j)\right| \nonumber \\&+\,\max _{1\le j\le n}\left| \sum _{i=1}^n Y_{ni}^{(3)}(j)\right| +\max _{1\le j\le n}\left| \sum _{i=1}^n Y_{ni}^{(4)}(j)\right| \nonumber \\\doteq & {} J_1+J_2+J_3+J_4. \end{aligned}$$

(3.29)

To prove (3.28), it suffices to show $J_i\rightarrow 0~a.s.,~n\rightarrow \infty ,~i=1,2,3,4$. We first prove $J_1\rightarrow 0~a.s.$, $n\rightarrow \infty$. For each j, we know that $\{Y_{ni}^{(1)}(j),1\le i\le n,n\ge 1\}$ is still an array of rowwise WOD random variables. In view of $EX_i=0$, (3.26), (3.27) and $EX^2<\infty$, we get

$$\begin{aligned}&\max _{1\le j\le n}\left| \sum _{i=1}^n E Y_{ni}^{(1)}(j)\right| \\&\quad \le \max _{1\le j\le n}\sum _{i=1}^n\left[ E|X_{ni}(j)|I\left( |X_{ni}(j)|> n^{-\delta }(h(n))^{-\beta /4}\right) \right. \\&\qquad +\left. n^{-\delta }(h(n))^{-\beta /4}P\left( |X_{ni}(j)|> n^{-\delta }(h(n))^{-\beta /4}\right) \right] \\&\quad \le 2\max _{1\le j\le n}\sum _{i=1}^nE|X_{ni}(j)|I\left( |X_{ni}(j)|> n^{-\delta }(h(n))^{-\beta /4}\right) \\&\quad \le C\max _{1\le j\le n}n^{\delta }(h(n))^{\beta /4}\sum _{i=1}^nE|X_{ni}(j)|^2I\left( |X_{ni}(j)|> n^{-\delta }(h(n))^{\beta }\right) \\&\quad \le Cn^{\delta }(h(n))^{\beta /4}\cdot \max _{1\le j\le n}\sum _{i=1}^na_{ni}(z_j)\cdot \max _{1\le i,j\le n}a_{ni}(z_j)\cdot EX^2\\&\quad \le Cn^{\delta -\alpha }(h(n))^{-3\beta /4}EX^2\rightarrow 0,~n\rightarrow \infty . \end{aligned}$$

Hence, for all n large enough, $\max \limits _{1\le j\le n}\left| \sum \nolimits _{i=1}^n E Y_{ni}^{(1)}(j)\right| <\frac{\varepsilon }{2}$. Applying Markov’s inequality and Rosenthal-type inequality, and taking

$$\begin{aligned} q>\max \left\{ \frac{2(\delta +1)-\alpha }{\delta },\frac{4}{\alpha },\frac{2}{\beta },2\right\} , \end{aligned}$$

we have

$$\begin{aligned}&\sum _{n=1}^\infty P\left( \max _{1\le j\le n}\left| \sum _{i=1}^n Y_{ni}^{(1)}(j)\right|>\varepsilon \right) \nonumber \\&\quad \le C\sum _{n=1}^\infty P\left( \max _{1\le j\le n}\left| \sum _{i=1}^n\left( Y_{ni}^{(1)}(j)-E Y_{ni}^{(1)}(j)\right) \right|>\frac{\varepsilon }{2}\right) \nonumber \\&\quad \le C\sum _{n=1}^\infty \sum _{j=1}^nP\left( \left| \sum _{i=1}^n\left( Y_{ni}^{(1)}(j)-E Y_{ni}^{(1)}(j)\right) \right| >\frac{\varepsilon }{2}\right) \nonumber \\&\quad \le C\sum _{n=1}^\infty \sum _{j=1}^n\left[ \sum _{i=1}^nE\left| Y_{ni}^{(1)}(j)\right| ^q+h(n)\sum _{i=1}^n\left( E\left| Y_{ni}^{(1)}(j)\right| ^2\right) ^{q/2}\right] \nonumber \\&\quad \doteq J_{11}+J_{12}. \end{aligned}$$

(3.30)

Note that

$$\begin{aligned} J_{11}\le & {} \sum _{n=1}^\infty \sum _{j=1}^n\sum _{i=1}^n\left[ n^{-\delta q}(h(n))^{-\frac{\beta q}{4}}P\left( |X_{ni}(j)|>n^{-\delta }(h(n))^{-\frac{\beta }{4}}\right) \right. \nonumber \\&+\,\left. E|X_{ni}(j)|^qI\left( |X_{ni}(j)|\le n^{-\delta }(h(n))^{-\frac{\beta }{4}}\right) \right] \nonumber \\\le & {} C\sum _{n=1}^\infty \sum _{j=1}^n\sum _{i=1}^n n^{-\delta (q-2)}(h(n))^{-\beta (q-2)/4}E|X_{ni}(j)|^2\nonumber \\\le & {} CEX^2\sum _{n=1}^\infty n^{1-\alpha -\delta (q-2)}(h(n))^{-\beta (q+2)/4}<\infty , \end{aligned}$$

(3.31)

and

$$\begin{aligned} J_{12}\le & {} C\sum _{n=1}^\infty \sum _{j=1}^nh(n)\left( \sum _{i=1}^n E|X_{ni}(j)|^2\right) ^{q/2}\nonumber \\\le & {} C(EX^2)^{q/2}\sum _{n=1}^\infty n^{1-\alpha q/2}(h(n))^{1-\beta q/2}<\infty . \end{aligned}$$

(3.32)

We can see that $J_1\rightarrow 0~a.s.$, $n\rightarrow \infty$ by (3.30)–(3.32) and the Borel–Cantelli Lemma.

Next we turn to estimate $J_2$. It follows from (3.27) that

$$\begin{aligned} \max _{1\le j\le n}\left| \sum _{i=1}^n Y_{ni}^{(2)}(j)\right|\le & {} C\max _{1\le j\le n}\sum _{i=1}^n \left| X_{ni}(j)\right| I\left( \left| X_{ni}(j)\right| \ge \frac{\varepsilon }{N}(h(n))^{-\beta /4}\right) \nonumber \\\le & {} Cn^{-\alpha }(h(n))^{-\beta }\sum _{i=1}^n|X_i|I\left( \left| X_{i}\right| \ge Cn^{\alpha }(h(n))^{\beta }(h(n))^{-\beta /4}\right) \nonumber \\\le & {} Cn^{-\alpha }(h(n))^{-\beta }\sum _{i=1}^n|X_i|I(\left| X_{i}\right| \ge Ci^{\alpha }). \end{aligned}$$

(3.33)

Hence, to prove $J_2\rightarrow 0~a.s.,~n\rightarrow \infty$, we only need to show

$$\begin{aligned} \sum _{i=1}^{\infty } i^{-\alpha }(h(i))^{-\beta }|X_i|I(\left| X_{i}\right| \ge Ci^{\alpha })<\infty ~a.s.. \end{aligned}$$

(3.34)

It can be checked by $\sum \nolimits _{i=1}^{n} i^{-\alpha }(h(i))^{-\beta }=O(n^{\alpha })$ and $EX^2<\infty$ that

$$\begin{aligned}&\sum _{i=1}^{\infty } i^{-\alpha }(h(i))^{-\beta }E|X_i|I(\left| X_{i}\right| \ge Ci^{\alpha })\nonumber \\&\quad \le C\sum _{i=1}^{\infty } i^{-\alpha }(h(i))^{-\beta }\sum _{n=i}^{\infty }E|X|I(Cn^{\alpha }\le |X|<C(n+1)^{\alpha })\nonumber \\&\quad \le C\sum _{n=1}^{\infty }n^{\alpha }E|X|I(Cn^{\alpha }\le |X|<C(n+1)^{\alpha }),\nonumber \\&\quad \le CEX^2<\infty , \end{aligned}$$

(3.35)

which implies that (3.34) holds. Consequently, according to (3.33), (3.34) and Kronecker’s lemma, $J_2\rightarrow 0~a.s.$, $n\rightarrow \infty$.

From the definition of $Y_{ni}^{(3)}(j)$, we know that

$$\begin{aligned} 0\le Y_{ni}^{(3)}(j)<\frac{\varepsilon }{N}(h(n))^{-\beta /4}-n^{-\delta }(h(n))^{-\beta /4}<\frac{\varepsilon }{N}. \end{aligned}$$

Therefore, by taking $N>\max \left\{ \frac{2}{\alpha -2\delta },\frac{2}{\beta }\right\}$, we have

$$\begin{aligned}&\sum _{n=1}^{\infty }P\left( \max _{1\le j\le n}\left| \sum _{i=1}^n Y_{ni}^{(3)}(j)\right| >\varepsilon \right) \\&\quad \le \sum _{n=1}^{\infty }\sum _{j=1}^nP\left( \text {there are at least N's nonzero}~Y_{ni}^{(3)}(j)\right) \\&\quad \le \sum _{n=1}^{\infty }\sum _{j=1}^n\sum _{1\le k_1<\cdots<k_N \le n}\\&\qquad P\left( X_{n,k_1}(j)\ge n^{-\delta }(h(n))^{-\beta /4},\ldots ,X_{n,k_N}(j)\ge n^{-\delta }(h(n))^{-\beta /4}\right) \\&\quad \le \sum _{n=1}^{\infty }\sum _{j=1}^n\sum _{1\le k_1<\cdots<k_N\le n}h(n)\prod _{i=1}^NP\left( X_{n,k_i}(j)\ge n^{-\delta }(h(n))^{-\beta /4}\right) \\&\quad \le \sum _{n=1}^{\infty }\sum _{j=1}^nh(n)\left( \sum _{i=1}^nP\left( |X_{ni}(j)|\ge n^{-\delta }(h(n))^{-\beta /4}\right) \right) ^N\\&\quad \le \sum _{n=1}^{\infty }\sum _{j=1}^nh(n)\left( \sum _{i=1}^nn^{2\delta }(h(n))^{\beta /2}E|X_{ni}(j)|^2\right) ^N\\&\quad \le C(EX^2)^N\sum _{n=1}^{\infty }n^{1-(\alpha -2\delta )N}(h(n))^{1-\beta N/2}<\infty . \end{aligned}$$

Hence, from the Borel–Cantelli lemma, we can obtain $J_3\rightarrow 0~a.s.$ $n\rightarrow \infty$. Note that

$$\begin{aligned} -\frac{\varepsilon }{N}<-\frac{\varepsilon }{N}(h(n))^{-\beta /4}+n^{-\delta /4}(h(n))^{-\beta }<Y_{ni}^{(4)}(j)\le 0 . \end{aligned}$$

Similar to the proof of $J_3$, we have $J_4\rightarrow 0~a.s.$ $n\rightarrow \infty$. This completes the proof of lemma. $\square$