Weak Law of Large Numbers and Mean Convergence for Weighted Sums of Multidimensional Arrays Under Gut’s Condition

Abstract

A new concept of weight-array-based stochastic domination in the Cesàro sense is developed. Under Gut’s condition, we establish the weak law of large numbers and ${\mathcal{L}}_r$-convergence for weighted sums of multidimensional arrays of pairwise negatively quadrant dependent random variables. These results improve and extend corresponding known results in the literature.

1. Introduction

The weak law of large numbers ensures that the sample mean converges in probability to the expected value, forming the foundation of consistent estimation. Under the independent and identically distributed (i.i.d. for short) setting, classical results are systematically presented in [1], and may be outlined as follows.

Theorem 1.

Let $\{X_n,n\ge 1\}$ be a sequence of i.i.d. random variables. Then as $n\to \infty$,

$$\begin{array}{c}\hfill {\displaystyle \frac{{\sum }_{i=1}^n\left(X_i-\mathbb{E}\left[X{1}_{\left\{\right|X|\le n\}}\right]\right)}{n}}\stackrel{\mathbb{P}}{\to }0\mathit{if}\mathit{and}\mathit{only}\mathit{if}n\mathbb{P}\left(\right|X_1|>n)\to 0.\end{array}$$

Let $0<p\le 1$ and $\mathcal{l}(·)$ be a slowly varying function. Ref. [2] provided an extension of Theorem 1 to a more general case, as described below.

Theorem 2.

Let $\{X_n,n\ge 1\}$ be a sequence of i.i.d. random variables. Then as $n\to \infty$,

$$\begin{array}{c}\hfill {\displaystyle \frac{{\sum }_{i=1}^n\left(X_i-\mathbb{E}\left[X{1}_{\left\{\right|X|\le b_n\}}\right]\right)}{n^{1/p}\mathcal{l}\left(n\right)}}\stackrel{\mathbb{P}}{\to }0\mathit{if}\mathit{and}\mathit{only}\mathit{if}n\mathbb{P}\left(\right|X_1|>n^{1/p}\mathcal{l}\left(n\right))\to 0.\end{array}$$

The Gut’s law stated in Theorem 2 was generalized by [3] to cover negatively associated (NA for short) random variables as well as the exponent regime $1\le p<2$. Further extensions can be found in [4,5,6,7,8,9] for instance. Very recently, ref. [10] made the first extension of Gut’s law to ${\mathcal{L}}_r$-convergence ($0<r<p,1\le p<2$) concerning maximum weighted sums of NA random variables, assuming that the weight sequence obeys ${\sum }_{i=1}^n{a}_{ni}^2=O\left(n\right)$.

A further approach to extending the weak law of large numbers consists of deriving analogous results for d-dimensional index arrays. Denote by ${\mathbb{Z}}_{+}^d$ the set of d-dimensional lattice points with positive integer coordinates, where $d\ge 1$. Define a componentwise partial order “⪯” on ${\mathbb{Z}}_{+}^d$; i.e., for $\mathbf{i}=(i_1,\cdots ,i_d)$ and $\mathbf{n}=(n_1,\cdots ,n_d)$, $\mathbf{i}⪯\mathbf{n}$ (or $\mathbf{n}⪰\mathbf{i}$) implies that $i_j\le n_j$ for each $1\le j\le d$. Denote $\left|\mathbf{n}\right|={\prod }_{j=1}^d{n}_j$. Consider a d-dimensional random field $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$. Ref. [11] studied an array of pairwise independent random variables in d dimensions under stochastic domination by X, and obtained that the condition $\mathbb{E}\left|X\right|<\infty$ yields

$$\begin{array}{c}\hfill {\displaystyle \frac{{\sum }_{\mathbf{i}⪯\mathbf{n}}(X_{\mathbf{i}}-\mathbb{E}\left[X_{\mathbf{i}}\right])}{\left|\mathbf{n}\right|}}\stackrel{{\mathcal{L}}_1}{\to }0as\left|\mathbf{n}\right|\to \infty .\end{array}$$

Ref. [12] established the ${\mathcal{L}}_r$ ($0<r<p$) convergence for multidimensional arrays of pairwise negatively quadrant dependent (NQD for short) random variables that are stochastically dominated in the Cesàro sense by a random variable X. They also relaxed the moment condition to $n\mathbb{P}\left(\right|X|>n^{1/p})\to 0$ as $n\to \infty$.

In many applications, however, the observations are not identically distributed and are influenced by a non-uniform weight field $\left\{a_{\mathbf{n},\mathbf{i}}\right\}$. For example, consider a two-dimensional random field $\left\{X_{i,j}\right\}$ representing daily rainfall measurements at grid points $(i,j)$. The weights $a_{\mathbf{n},\mathbf{i}}$ can model directional or distance-based attenuation of sensor signals. The total impact on a distributed system is a weighted sum ${\sum }_{i,j}{a}_{n_1,n_2,i,j}{X}_{i,j}$. Under this circumstance, traditional unweighted results fail to capture the influence of non-homogeneous weights. Moreover, Gut’s condition $n\mathbb{P}\left(\right|X|>{\Delta }_n)\to 0$ with ${\Delta }_n=n^{1/p}\mathcal{l}\left(n\right)$ would substantially weaken the tail requirements on the dominating variable X, making the law of large numbers applicable to a wider class of heavy-tailed distributions where $\mathcal{l}\left(n\right)\to \infty$ (see Example 1, where the advantage of Gut’s condition is demonstrated).

The main contribution of this work is twofold: (i) extending the results of [10] from NA random variables to multidimensional arrays of pairwise NQD random variables under Gut’s condition, and (ii) improving the ${\mathcal{L}}_r$-convergence and weak law results of [12] from partial sums to weighted sums.

To end this section, let us recall some concepts which are indispensable to illustrate our main results. Ref. [13] proposed the following concept of stochastic domination in the Cesàro sense.

Definition 1.

A d-dimensional random field $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is stochastically dominated in the Cesàro sense by X if, for some $M>0$, the inequality

$${\displaystyle \frac{1}{\left|\mathbf{n}\right|}}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|X_{\mathbf{i}}|>x)\le M\mathbb{P}\left(\right|X|>x)$$

holds for every $x\ge 0$ and $\mathbf{n}\in {\mathbb{Z}}_{+}^d$.

The Cesàro-type stochastic domination introduced above is strictly weaker than the standard stochastic domination, yet it is insufficient for deriving our main results. To address this issue, we propose a new notion of stochastic domination in the Cesàro sense relative to a given array.

Definition 2.

Let $\mathbf{1}=(1,\dots ,1)\in {\mathbb{Z}}_{+}^d$. A random field $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is said to be stochastically dominated in the Cesàro sense by a random variable X with respect to the array $\{a_{\mathbf{n},\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ if there exists a finite positive constant M such that for all $x\ge 0$ and $\mathbf{n}\in {\mathbb{Z}}_{+}^d$,

$$\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>x)\le M\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}X|>x).$$

Remark 1.

Observe that when $a_{\mathbf{n},\mathbf{i}}\equiv 1$ for every $\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d$, Definition 2 reduces to the original Cesàro-type stochastic domination in Definition 1. Moreover, it is worth noting that the usual stochastic domination (i.e., $\mathbb{P}\left(\right|X_{\mathbf{i}}|>x)\le M\mathbb{P}\left(\right|X|>x)$ for all $\mathbf{i}$ and x) also implies the Cesàro-type stochastic domination in Definition 2 with respect to any weight array $\left\{a_{\mathbf{n},\mathbf{i}}\right\}$. Indeed, if $\mathbb{P}\left(\right|X_{\mathbf{i}}|>x)\le M\mathbb{P}\left(\right|X|>x)$ holds for each $\mathbf{i}$, then summing over $\mathbf{i}⪯\mathbf{n}$ gives ${\sum }_{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>x)\le M{\sum }_{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}X|>x)$.

The notion of pairwise NQD random variables was introduced by [14] as follows.

Definition 3.

Two random variables X and Y are called NQD if

$$\mathbb{P}(X\le x,Y\le y)\le \mathbb{P}(X\le x)\mathbb{P}(Y\le y).$$

A sequence $\{X_n,n\ge 1\}$ is said to be pairwise NQD if every pair of variables in the sequence satisfies this NQD property.

It is worth noting that negatively orthant dependent, negatively superadditive dependent, NA, or pairwise independent sequences all belong to the class of pairwise NQD variables. Hence, this class is quite broad, and investigating its limit properties is of great significance. Extensive limit theorems have been established for pairwise NQD random variables, including the Kolmogorov strong law of large numbers for identically distributed sequences by [15]; the Marcinkiewicz-type strong law under suitable conditions by [16]; strong stability for Jamison-type weighted product sums by [17]; maximal moment inequalities by [18]; limit theorems by [19]; the Hájek–Rényi-type inequality by [20]; strong laws under normalizing sequences by [21]; and complete moment convergence for weighted sums by [22], among others. Ref. [12] generalized the pairwise NQD concept to the multidimensional array setting.

Definition 4.

A d-dimensional random field $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is termed pairwise NQD if for every pair of distinct indices $\mathbf{i}\ne \mathbf{j}$ and any real numbers $x,y$,

$$\begin{array}{c}\hfill \mathbb{P}(X_{\mathbf{i}}\le x,X_{\mathbf{j}}\le y)\le \mathbb{P}(X_{\mathbf{i}}\le x)\mathbb{P}(X_{\mathbf{j}}\le y).\end{array}$$

According to [12], this dependence structure appears in many natural examples, including the cone measure and the volume measure on the ball of ${\mathcal{l}}_r^n$.

We also recall the definition of a slowly varying function. A positive measurable function $\mathcal{l}(·)$ on ${\mathbb{R}}^{+}$ is called slowly varying if, for any $t>0$,

$$\begin{array}{c}\hfill \underset{x\to \infty }{\lim }{\displaystyle \frac{\mathcal{l}\left(tx\right)}{\mathcal{l}\left(x\right)}}=1.\end{array}$$

Clearly, many functions are slowly varying, e.g., ${\log }^{r}x$, ${\log }^s\log x$, and ${\displaystyle \frac{{\log }^{t}x}{\log \log x}}$ with arbitrary $r,s,t>0$; here, $\log x=\ln \max (x,e)$ and $\ln x$ denotes the natural logarithm.

Finally, we establish the following notational conventions. C denotes a generic positive constant that may vary across occurrences. $a_n=o\left(b_n\right)$ means ${\lim }_{n\to \infty }{a}_n/b_n=0$; $a_n\asymp b_n$ indicates $a_n=O\left(b_n\right)$ and $b_n=O\left(a_n\right)$. Moreover, $x^{+}=x{1}_{\{x\ge 0\}}$, $x^{-}=-x{1}_{\{x<0\}}$, with $1_A$ being the indicator of event A.

2. Main Results and an Application

2.1. Main Results

Let $\{b_n,n\ge 1\}$ be a sequence of positive reals tending to infinity. Our first result provides a general weak law of large numbers for weighted sums of pairwise NQD random fields.

Theorem 3.

Suppose $\{a_{\mathbf{n},\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is an array of constants and $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is a pairwise NQD random field. If as $|\mathbf{n}|\to \infty$,

• (i) ${\sum }_{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b_{|\mathbf{n}|})\to 0$,
• (ii) $b_{|\mathbf{n}|}^{-2}{\sum }_{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^2{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{|\mathbf{n}|}\}}]\to 0$,
• then we get

$$\begin{array}{c}\hfill {\displaystyle \frac{{\sum }_{\mathbf{i}⪯\mathbf{n}}(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{|\mathbf{n}|}\}}\right])}{b_{|\mathbf{n}|}}}\stackrel{\mathbb{P}}{\to }0as|\mathbf{n}|\to \infty .\end{array}$$

Applying Theorem 3, we derive the following Kolmogorov-type weak law of large numbers for weighted sums of pairwise NQD random fields.

Corollary 1.

Let $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ be a pairwise NQD random field that is stochastically dominated in the Cesàro sense by a random variable X with respect to the constant array $\{a_{\mathbf{n},\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$, where ${\sum }_{\mathbf{i}⪯\mathbf{n}}|a_{\mathbf{n},\mathbf{i}}{|}^q=O(|\mathbf{n}|)$ for some $q>1$. If $n\mathbb{P}(|X|>n)\to 0$ as $n\to \infty$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{{\sum }_{\mathbf{i}⪯\mathbf{n}}(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le |\mathbf{n}|\}}\right])}{|\mathbf{n}|}}\stackrel{\mathbb{P}}{\to }0\mathrm{as}|\mathbf{n}|\to \infty .\end{array}$$

The next theorem establishes the ${\mathcal{L}}_r$-convergence for weighted sums of multidimensional arrays of pairwise NQD random variables under Gut’s condition.

Theorem 4.

Let $0<r<p$, $1\le p<2$, and set ${\Delta }_n=n^{1/p}\mathcal{l}\left(n\right)$ with $\mathcal{l}(·)$ slowly varying. Consider a pairwise NQD random field $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ that is stochastically dominated in the Cesàro sense by X with respect to the constant array $\{a_{\mathbf{n},\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$, where ${\sum }_{\mathbf{i}⪯\mathbf{n}}|a_{\mathbf{n},\mathbf{i}}{|}^q=O(|\mathbf{n}|)$ for some $q>p$. Suppose that $n\mathbb{P}(|X|>{\Delta }_n)\to 0$ as $n\to \infty$. Then, for $1<p<2$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}(X_{\mathbf{i}}-\mathbb{E}\left[X_{\mathbf{i}}\right])\right|\stackrel{{\mathcal{L}}_r}{\to }0\mathrm{as}\left|\mathbf{n}\right|\to \infty ,\end{array}$$

provided that $\mathbb{E}|X_{\mathbf{i}}|<\infty$ for all $\mathbf{1}⪯\mathbf{i}⪯\mathbf{n}$; and when $p=1$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {\Delta }_{\left|\mathbf{n}\right|}\}}\right])\right|\stackrel{{\mathcal{L}}_r}{\to }0\mathrm{as}\left|\mathbf{n}\right|\to \infty .\end{array}$$

Remark 2.

If we take $a_{\mathbf{n},\mathbf{i}}\equiv 1$ for all $\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d$ and set $\mathcal{l}(·)=1$, then Theorem 4 reduces to Theorem 3.2 of [12]. Thus, our result generalizes that of [12].

Remark 3.

If the dimension $d=1$ and the stochastic domination in the Cesàro sense with respect to the array $\{a_{\mathbf{n},\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is replaced with a restrictive assumption, i.e., stochastic domination (which implies the former; cf. Remark 1), Theorem 4 still extends Theorems 2.1 and 2.2 of [10] from NA setting to pairwise NQD setting. Hence, our result also improves and extends the corresponding ones of [10] to the case of multidimensional array.

We now construct an example illustrating that the condition $n\mathbb{P}\left(\right|X|>{\Delta }_n)\to 0$ in Theorem 4 is weaker than the condition $n\mathbb{P}\left(\right|X|>n^{1/p})\to 0$ used in [12].

Example 1.

Consider a random field $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ satisfying

$$\begin{array}{c}\hfill \mathbb{P}(X_{\mathbf{n}}=0)=1-{\displaystyle \frac{1}{\left|\mathbf{n}\right|}},\mathbb{P}(X_{\mathbf{n}}={\left|\mathbf{n}\right|}^{1/p})={\displaystyle \frac{1}{\left|\mathbf{n}\right|}}.\end{array}$$

As shown in Example 1 of [23], this field is stochastically dominated in the Cesàro sense by a random variable X with $\mathbb{P}(X>x)=1$ for $x<1$ and $\mathbb{P}(X>x)={\left(\lceil x^p\rceil \right)}^{-1}$ for $x\ge 1$ (where $\lceil ·\rceil$ is the ceiling function). Consequently, we have $n\mathbb{P}\left(\right|X|>n^{1/p})\to 1$ while $n\mathbb{P}\left(\right|X|>{\Delta }_n)\to 0$ whenever $\mathcal{l}\left(n\right)\to \infty$ as $n\to \infty$.

For the case $0<p<1$, we can establish the following result on the maximum of weighted sums, which imposes no dependence restrictions on the random field.

Theorem 5.

Let $0<r<p<1$ and ${\Delta }_n=n^{1/p}\mathcal{l}\left(n\right)$ with $\mathcal{l}(·)$ slowly varying. Suppose $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is a random field (without any dependence assumptions) that is stochastically dominated in the Cesàro sense by X with respect to the constant array $\{a_{\mathbf{n},\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$, where ${\sum }_{\mathbf{i}⪯\mathbf{n}}|a_{\mathbf{n},\mathbf{i}}{|}^q=O\left(\right|\mathbf{n}\left|\right)$ for some $q>p$. If $n\mathbb{P}\left(\right|X|>{\Delta }_n)\to 0$ as $n\to \infty$, then

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\underset{\mathbf{k}⪯\mathbf{n}}{\max }\left|\sum _{\mathbf{i}⪯\mathbf{k}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}\right|\stackrel{{\mathcal{L}}_r}{\to }0\mathrm{as}\left|\mathbf{n}\right|\to \infty .\end{array}$$

To further demonstrate the influence and practical power of our main results, we provide a concrete illustration as follows.

Example 2.

Consider the cone measure on the unit ball of ${\mathcal{l}}_p^n$, which induces a pairwise NQD coordinate sequence $\{X_{\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n}\}$ as noted by [12]. In high-dimensional geometry, certain natural quantities of interest involve weighted sums of the coordinates, e.g., ${\sum }_{\mathbf{i}⪯\mathbf{n}}{w}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}$ with weights $w_{\mathbf{n},\mathbf{i}}$ such that ${\sum }_{\mathbf{i}⪯\mathbf{n}}|w_{\mathbf{n},\mathbf{i}}{|}^q=O\left(\right|\mathbf{n}\left|\right)$ for some $q>p$.

Suppose the coordinate marginals are stochastically dominated in the Cesàro sense by a heavy-tailed random variable X satisfying $n\mathbb{P}\left(\right|X|>n^{1/p})\nrightarrow 0$, while for some slowly varying function $\mathcal{l}(·)\to \infty$, we have $n\mathbb{P}\left(\right|X|>n^{1/p}\mathcal{l}\left(n\right))\to 0$. Such tail behavior can be easily achieved; see Example 1 for an example. Then Theorem 4 guarantees the ${\mathcal{L}}_r$-convergence of the normalized weighted sums, whereas the results of [12] are inapplicable. This application highlights how the combination of weighted sums, Cesàro-type domination, and Gut’s condition overcomes limitations of earlier results.

2.2. Application to Spatial Kernel Regression

To illustrate the applicability of Theorem 4, consider the Nadaraya–Watson surface estimator for a random field on a 2-dimensional grid. Let $\{Y_{\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n}\}$ with $\mathbf{i}=(i_1,i_2)$ be modeled as

$$Y_{\mathbf{i}}={m(\mathbf{i}/|\mathbf{n}|}^{1/2})+{\epsilon }_{\mathbf{i}},$$

where $m\in C^2\left({[0,1]}^2\right)$ (i.e., m has continuous partial derivatives up to order two) and $\left\{{\epsilon }_{\mathbf{i}}\right\}$ is a zero-mean pairwise NQD field admitting Cesàro-type stochastic domination by X.

The Nadaraya–Watson estimator of $m\left(\mathbf{x}\right)$ at a fixed point $\mathbf{x}=(x_1,x_2)\in {(0,1)}^2$ can be written as ([24], Equation (5.23))

$${\widehat{m}}_{\mathbf{n}}\left(\mathbf{x}\right)={\displaystyle \frac{{\sum }_{\mathbf{i}⪯\mathbf{n}}K\left(\frac{i_1/{\left|\mathbf{n}\right|}^{1/2}-x_1}{b_{\left|\mathbf{n}\right|}}\right)K\left(\frac{i_2/{\left|\mathbf{n}\right|}^{1/2}-x_2}{b_{\left|\mathbf{n}\right|}}\right)Y_{\mathbf{i}}}{{\sum }_{\mathbf{i}⪯\mathbf{n}}K\left(\frac{i_1/{\left|\mathbf{n}\right|}^{1/2}-x_1}{b_{\left|\mathbf{n}\right|}}\right)K\left(\frac{i_2/{\left|\mathbf{n}\right|}^{1/2}-x_2}{b_{\left|\mathbf{n}\right|}}\right)}},$$

where K is a bounded symmetric density with support $[-1,1]$, and the bandwidth satisfies $b_{\left|\mathbf{n}\right|}\to 0$, ${\left|\mathbf{n}\right|}^{1/2}{b}_{\left|\mathbf{n}\right|}\to \infty$. The denominator is asymptotically equivalent to $\left|\mathbf{n}\right|b_{\left|\mathbf{n}\right|}^2$ up to a constant, so the estimation error admits the decomposition

$${\widehat{m}}_{\mathbf{n}}\left(\mathbf{x}\right)-m\left(\mathbf{x}\right)=B_{\mathbf{n}}\left(\mathbf{x}\right)+\sum _{\mathbf{i}⪯\mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}{\epsilon }_{\mathbf{i}},$$

with $B_{\mathbf{n}}\left(\mathbf{x}\right)=O\left(b_{\left|\mathbf{n}\right|}^2\right)$ and effective weights

$$a_{\mathbf{n},\mathbf{i}}\asymp {\displaystyle \frac{1}{\left|\mathbf{n}\right|b_{\left|\mathbf{n}\right|}^2}}K\left({\displaystyle \frac{i_1/{\left|\mathbf{n}\right|}^{1/2}-x_1}{b_{\left|\mathbf{n}\right|}}}\right)K\left({\displaystyle \frac{i_2/{\left|\mathbf{n}\right|}^{1/2}-x_2}{b_{\left|\mathbf{n}\right|}}}\right).$$

To apply Theorem 4, set ${\Delta }_n=n^{1/p}\mathcal{l}\left(n\right)$ with $1<p<2$, and rewrite the stochastic term as

$$\sum _{\mathbf{i}⪯\mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}{\epsilon }_{\mathbf{i}}={\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\sum _{\mathbf{i}⪯\mathbf{n}}{\tilde{a}}_{\mathbf{n},\mathbf{i}}{\epsilon }_{\mathbf{i}},{\tilde{a}}_{\mathbf{n},\mathbf{i}}={\displaystyle \frac{{\Delta }_{\left|\mathbf{n}\right|}}{\left|\mathbf{n}\right|b_{\left|\mathbf{n}\right|}^2}}K\left({\displaystyle \frac{i_1/{\left|\mathbf{n}\right|}^{1/2}-x_1}{b_{\left|\mathbf{n}\right|}}}\right)K\left({\displaystyle \frac{i_2/{\left|\mathbf{n}\right|}^{1/2}-x_2}{b_{\left|\mathbf{n}\right|}}}\right).$$

Due to the compact support of K, the sum contains only $O\left(\right|\mathbf{n}\left|b_{\left|\mathbf{n}\right|}^2\right)$ nonzero terms. Moreover, boundedness of K implies that each $|{\tilde{a}}_{\mathbf{n},\mathbf{i}}|$ satisfies $|{\tilde{a}}_{\mathbf{n},\mathbf{i}}|\asymp {\Delta }_{\left|\mathbf{n}\right|}/\left(\right|\mathbf{n}|b_{\left|\mathbf{n}\right|}^2)$. Consequently,

$$\sum _{\mathbf{i}⪯\mathbf{n}}|{\tilde{a}}_{\mathbf{n},\mathbf{i}}{|}^q\asymp \left|\mathbf{n}\right|b_{\left|\mathbf{n}\right|}^2{\left({\displaystyle \frac{{\Delta }_{\left|\mathbf{n}\right|}}{\left|\mathbf{n}\right|b_{\left|\mathbf{n}\right|}^2}}\right)}^q={\Delta }_{\left|\mathbf{n}\right|}^q{\left|\mathbf{n}\right|}^{1-q}{b}_{\left|\mathbf{n}\right|}^{2-2q}.$$

Choose the bandwidth as

$$b_{\left|\mathbf{n}\right|}\asymp {\left({\displaystyle \frac{{\Delta }_{\left|\mathbf{n}\right|}}{\left|\mathbf{n}\right|}}\right)}^{1/2}.$$

Then, for any $q>p$, the weight condition ${\sum }_{\mathbf{i}⪯\mathbf{n}}|{\tilde{a}}_{\mathbf{n},\mathbf{i}}{|}^q=O\left(\right|\mathbf{n}\left|\right)$ is fulfilled. Moreover, $b_{\left|\mathbf{n}\right|}\sim {\left|\mathbf{n}\right|}^{{\displaystyle \frac{1}{2p}}-{\displaystyle \frac{1}{2}}}\mathcal{l}{\left(\right|\mathbf{n}\left|\right)}^{1/2}\to 0$ and the bias satisfies $B_{\mathbf{n}}\left(\mathbf{x}\right)=O\left(b_{\left|\mathbf{n}\right|}^2\right)=O({\Delta }_{\left|\mathbf{n}\right|}/\left|\mathbf{n}\right|)\to 0$.

Now assume that the dominating variable X satisfies Gut’s condition $n\mathbb{P}\left(\right|X|>{\Delta }_n)\to 0$. Since $\left\{{\epsilon }_{\mathbf{i}}\right\}$ is pairwise NQD and Cesàro-dominated by X, Theorem 4 directly yields

$$\sum _{\mathbf{i}⪯\mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}{\epsilon }_{\mathbf{i}}={\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\sum _{\mathbf{i}⪯\mathbf{n}}{\tilde{a}}_{\mathbf{n},\mathbf{i}}{\epsilon }_{\mathbf{i}}\stackrel{{\mathcal{L}}_r}{\to }0as\left|\mathbf{n}\right|\to \infty ,0<r<p.$$

Consequently,

$${\widehat{m}}_{\mathbf{n}}\left(\mathbf{x}\right)\stackrel{{\mathcal{L}}_r}{\to }m\left(\mathbf{x}\right).$$

Remark 4.

In the common situation, where the slowly varying function $\mathcal{l}(·)$ is constant (i.e., $\mathcal{l}\left(n\right)\equiv 1$), so that ${\Delta }_n=n^{1/p}$ with $1<p<2$. Then the chosen bandwidth becomes

$$b_{\left|\mathbf{n}\right|}\asymp {\left({\displaystyle \frac{{\Delta }_{\left|\mathbf{n}\right|}}{\left|\mathbf{n}\right|}}\right)}^{1/2}={\left|\mathbf{n}\right|}^{{\displaystyle \frac{1}{2p}}-{\displaystyle \frac{1}{2}}}.$$

In two-dimensional nonparametric regression with a twice-differentiable regression function $m\in C^2\left({[0,1]}^2\right)$, the optimal bandwidth that minimizes the mean integrated squared error is $b_{\mathrm{opt}}\asymp {\left|\mathbf{n}\right|}^{-1/6}$ (see, e.g., Chapter 2 in [25]). This rate is achieved by taking $p=3/2$, which lies in the admissible interval $(1,2)$. For this choice, Gut’s condition reduces to $n\mathbb{P}\left(\right|X|>n^{2/3})\to 0$, a natural tail condition on the dominating variable X which is weaker than requiring finite pth moments. Hence, our theoretical assumptions cover practically relevant bandwidth selections.

3. Some Important Lemmas

Prior to presenting the proofs of our main theorems, we need the following auxiliary results.

Lemma 1.

If X and Y are NQD random variables, then for any nondecreasing (or nonincreasing) functions f and g, the transformed variables $f\left(X\right)$ and $g\left(Y\right)$ remain NQD.

Lemma 2.

Let $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ be a pairwise NQD random field and $\{f_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ a family of monotone functions (all nondecreasing or all nonincreasing). Then the transformed field $\{f_{\mathbf{n}}\left(X_{\mathbf{n}}\right),\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is also pairwise NQD.

Proof. 

For any two distinct indices $\mathbf{i}\ne \mathbf{j}$, the random variables $X_{\mathbf{i}}$ and $X_{\mathbf{j}}$ are pairwise NQD by definition. Therefore, it follows from Lemma 1 that $f_{\mathbf{i}}\left(X_{\mathbf{i}}\right)$ and $f_{\mathbf{j}}\left(X_{\mathbf{j}}\right)$ are also pairwise NQD. This verifies the lemma immediately. □

Ref. [26] established the following von Bahr–Esseen-type inequality.

Lemma 3.

Let $\{X_n,n\ge 1\}$ be pairwise NQD random variables with zero means and finite p-th moments for some $1\le p\le 2$. Then there exists a constant $C>0$ depending only on p such that

$$\begin{array}{c}\hfill \mathbb{E}{\left|{\displaystyle \sum _{i=1}^n}{X}_i\right|}^p\le C{\displaystyle \sum _{i=1}^n}\mathbb{E}\left[\right|X_i{|}^p].\end{array}$$

Lemma 4.

Let $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ be a d-dimensional pairwise NQD random field with $\mathbb{E}{X}_{\mathbf{n}}=0$ and $\mathbb{E}|X_{\mathbf{n}}{|}^p<\infty$ for some $1\le p\le 2$. Then for some constant $C>0$ (depending only on p), the following inequality holds for every $\mathbf{n}\in {\mathbb{Z}}_{+}^d$:

$$\begin{array}{c}\hfill \mathbb{E}{\left|\sum _{\mathbf{i}⪯\mathbf{n}}{X}_{\mathbf{i}}\right|}^p\le C\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[\right|X_{\mathbf{i}}{|}^p].\end{array}$$

Proof. 

Write $\mathbf{n}=(n_1,\cdots ,n_d)$ and $\left|\mathbf{n}\right|={\prod }_{j=1}^d{n}_j$. Define the finite set of indices

$$\begin{array}{c}\hfill \mathcal{I}=\{\mathbf{i}=(i_1,\cdots ,i_d):1\le i_k\le n_k,1\le k\le d\}.\end{array}$$

Noting that there exists a bijection $\phi :\mathcal{I}\leftrightarrow \{1,2,\dots ,|\mathbf{n}\left|\right\}$, we define for each $1\le j\le \left|\mathbf{n}\right|$ that $Z_j=X_{{\phi }^{-1}\left(j\right)}$. Therefore, for every original index $\mathbf{i}\in \mathcal{I}$, there is a unique j such that $Z_j=X_{\mathbf{i}}$. Actually, the random variables are merely relabeled and thus the one-dimensional sequence $\{Z_j,1\le j\le |\mathbf{n}\left|\right\}$ is still pairwise NQD by the definition of NQD random variables. Hence, applying Lemma 3 and observing that $\phi$ is bijection, we can obtain

$$\begin{array}{c}\hfill \mathbb{E}{\left|\sum _{\mathbf{i}⪯\mathbf{n}}{X}_{\mathbf{i}}\right|}^p=\mathbb{E}{\left|{\displaystyle \sum _{j=1}^{\left|\mathbf{n}\right|}}{Z}_j\right|}^p\le C{\displaystyle \sum _{j=1}^{\left|\mathbf{n}\right|}}\mathbb{E}\left[\right|Z_j{|}^p]=C\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[\right|X_{\mathbf{i}}{|}^p].\end{array}$$

The proof is complete. □

Lemma 5.

Suppose $\{X_{\mathbf{n}},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is a random field that is stochastically dominated in the Cesàro sense by X with respect to the constant array $\{a_{\mathbf{n},\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$. Then for any positive constants a and b,

$$\begin{array}{ccc}& & \sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^a{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b\}}]\le M\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^a{1}_{\{|a_{ni}X|>b\}}];\hfill \\ & & \sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^a{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b\}}]\le M\sum _{\mathbf{i}⪯\mathbf{n}}\left[b^a\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>b)+\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^a{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le b\}}]\right].\hfill \end{array}$$

Proof. 

It follows from Definition 2 and the fact ${\mathbb{E}[|\zeta |}^a{1}_{\{|\zeta |>b\}}]=b^a\mathbb{P}(|\zeta |>b)+{\int }_{b^a}^{\infty }\mathbb{P}(|\zeta |>x^{1/a})dx$ that

$$\begin{array}{ccc}& & \sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^a{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b\}}]\hfill \\ & =& \sum _{\mathbf{i}⪯\mathbf{n}}\left[b^a\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b)+{\int }_{b^a}^{\infty }\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>x^{1/a})dx\right]\hfill \\ & =& b^a\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b)+{\int }_{b^a}^{\infty }\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>x^{1/a})dx\hfill \\ & \le & M{b}^a\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>b)+M{\int }_{b^a}^{\infty }\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>x^{1/a})dx\hfill \\ & =& M\sum _{\mathbf{i}⪯\mathbf{n}}\left[{\int }_0^{b^a}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>b)dx+{\int }_{b^a}^{\infty }\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>x^{1/a})dx\right]\hfill \\ & =& M\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^a{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|>b\}}].\hfill \end{array}$$

Moreover, by virtue of $a{\int }_0^b{x}^{a-1}\mathbb{P}(|\zeta |>x)dx=b^a\mathbb{P}(|\zeta |>b)+{\mathbb{E}[|\zeta |}^a{1}_{\{|\zeta |\le b\}}]$ and Definition 2 again, we have that

$$\begin{array}{ccc}& & \sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^a{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b\}}]\hfill \\ & \le & \sum _{\mathbf{i}⪯\mathbf{n}}a{\int }_0^b{x}^{a-1}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>x)dx\hfill \\ & =& a{\int }_0^b{x}^{a-1}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>x)dx\hfill \\ & \le & M\sum _{\mathbf{i}⪯\mathbf{n}}a{\int }_0^b{x}^{a-1}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>x)dx\hfill \\ & =& M\sum _{\mathbf{i}⪯\mathbf{n}}\left[b^a\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>b)+\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^a{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le b\}}]\right].\hfill \end{array}$$

The proof is completed. □

The last lemma can be found in Lemma 1.2 of [10] as follows.

Lemma 6.

Let $\mathcal{l}(·)$ be slowly varying. Assume (after possibly replacing $\mathcal{l}\left(x\right)$ by $C\mathcal{l}\left(x\right)$ with an appropriate C) that for some $a>0$ sufficiently large, the function $x^r\mathcal{l}\left(x\right)$ is strictly increasing on $[a,\infty )$ and $x^{-r}\mathcal{l}\left(x\right)$ is strictly decreasing on $[a,\infty )$. Then the following asymptotic equivalences hold.

(i) For $\alpha >-1$ and $\beta >0$, we have ${\sum }_{k=1}^n{k}^{\alpha }\mathcal{l}\left(k^{\beta }\right)\asymp n^{\alpha +1}\mathcal{l}\left(n^{\beta }\right)$.

(ii) For $\alpha <-1$ and $\beta >0$, we have ${\sum }_{k=n}^{\infty }{k}^{\alpha }\mathcal{l}\left(k^{\beta }\right)\asymp n^{\alpha +1}\mathcal{l}\left(n^{\beta }\right)$.

(iii) If $p,q>0$, and ${\Delta }_n=n^{1/p}\mathcal{l}\left(n\right)$, $n\ge 1$, then for all $k\ge 1$,

$${\Delta }_{k+1}^q-{\Delta }_k^q\le {\displaystyle \frac{2^{q/p}(p+q)}{p}}{k}^{q/p-1}{\mathcal{l}}^q\left(k\right).$$

4. Proofs

Before presenting the detailed proofs, we briefly comment on the key technical novelties that allow us to go beyond the existing frameworks.

∗Multidimensional NQD inequality via bijection. In Lemma 4, we reduce the moment estimate for a d-dimensional array to the one-dimensional von Bahr–Esseen inequality by simply relabeling the index set through a bijection $\varphi$. This elementary trick makes the whole proof essentially as simple as the one-dimensional case, while still valid for all $d\ge 1$.

∗Cesàro stochastic domination with weights. Lemma 5, built on Definition 2, provides the crucial bounds

$$\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^a{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b\}}]\le M\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[\right|a_{\mathbf{n},\mathbf{i}}X{|}^a{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}X|>b\}}],$$

and a similar bound for the truncated part. This allows us to replace the original weighted variables by those dominated by X, while keeping the weight structure intact. Consequently, the dependence on the specific distributions of $X_{\mathbf{i}}$ is eliminated, and only the tail of X matters.

With these points in mind, we now proceed to the rigorous proofs.

Proof of Theorem 3.

Since $a_{\mathbf{n},\mathbf{i}}=a_{\mathbf{n},\mathbf{i}}^{+}-a_{\mathbf{n},\mathbf{i}}^{-}$ for each $\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d$, one can first prove

$$\begin{array}{c}\hfill {\displaystyle \frac{{\sum }_{\mathbf{i}⪯\mathbf{n}}\left(a_{\mathbf{n},\mathbf{i}}^{+}{X}_{\mathbf{i}}-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}^{+}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}^{+}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}\right]\right)}{b_{\left|\mathbf{n}\right|}}}\stackrel{\mathbb{P}}{\to }0as\left|\mathbf{n}\right|\to \infty \end{array}$$

and

$$\begin{array}{c}\hfill {\displaystyle \frac{{\sum }_{\mathbf{i}⪯\mathbf{n}}\left(a_{\mathbf{n},\mathbf{i}}^{-}{X}_{\mathbf{i}}-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}^{-}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}^{-}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}\right]\right)}{b_{\left|\mathbf{n}\right|}}}\stackrel{\mathbb{P}}{\to }0as\left|\mathbf{n}\right|\to \infty ,\end{array}$$

which together with the fact that

$$\begin{array}{c}\hfill a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}=a_{\mathbf{n},\mathbf{i}}^{+}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}^{+}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}-a_{\mathbf{n},\mathbf{i}}^{-}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}^{-}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}\end{array}$$

yields that

$$\begin{array}{ccc}& & {\displaystyle \frac{{\sum }_{\mathbf{i}⪯\mathbf{n}}\left(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}\right]\right)}{b_{\left|\mathbf{n}\right|}}}\hfill \\ & =& {\displaystyle \frac{{\sum }_{\mathbf{i}⪯\mathbf{n}}\left(a_{\mathbf{n},\mathbf{i}}^{+}{X}_{\mathbf{i}}-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}^{+}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}^{+}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}\right]\right)}{b_{\left|\mathbf{n}\right|}}}\hfill \\ & & -{\displaystyle \frac{{\sum }_{\mathbf{i}⪯\mathbf{n}}\left(a_{\mathbf{n},\mathbf{i}}^{-}{X}_{\mathbf{i}}-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}^{-}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}^{-}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}\right]\right)}{b_{\left|\mathbf{n}\right|}}}\stackrel{\mathbb{P}}{\to }0as\left|\mathbf{n}\right|\to \infty .\hfill \end{array}$$

Since both $\left\{a_{\mathbf{n},\mathbf{i}}^{+}\right\}$ and $\left\{a_{\mathbf{n},\mathbf{i}}^{-}\right\}$ consist of non-negative coefficients, it is enough to establish the statement for non-negative coefficients $a_{\mathbf{n},\mathbf{i}}\ge 0$; the general case is then handled by applying the result to the positive and negative parts separately and taking the difference (the required conditions carry over because $a_{\mathbf{n},\mathbf{i}}^{\pm }\le \left|a_{\mathbf{n},\mathbf{i}}\right|$, and the pairwise NQD property is preserved under the nondecreasing transformations $x\mapsto a_{\mathbf{n},\mathbf{i}}^{\pm }x$, see Lemma 2). Consequently, we can assume, with no loss of generality, that $a_{\mathbf{n},\mathbf{i}}\ge 0$ holds for every $\mathbf{1}⪯\mathbf{i}⪯\mathbf{n}$ and $\mathbf{n}\in {\mathbb{Z}}_{+}^d$. For each $\mathbf{n}\in {\mathbb{Z}}_{+}^d$, denote

$$\begin{array}{c}\hfill Y_{\mathbf{n},\mathbf{i}}=-b_{\left|\mathbf{n}\right|}{1}_{\{a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}<-b_{\left|\mathbf{n}\right|}\}}+a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}+b_{\left|\mathbf{n}\right|}{1}_{\{a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}>b_{\left|\mathbf{n}\right|}\}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\end{array}$$

where $\mathbf{1}=(1,\dots ,1)\in {\mathbb{Z}}_{+}^d$, $\{a_{\mathbf{n},\mathbf{i}},\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is a field of constants and $\{b_n,n\ge 1\}$ is a sequence of positive numbers diverging to infinity. It is sufficient to demonstrate that for every $\epsilon >0$,

$$\begin{array}{c}\hfill \mathbb{P}\left(\left|\sum _{\mathbf{i}⪯\mathbf{n}}(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}-Y_{\mathbf{n},\mathbf{i}})\right|>\epsilon b_{\left|\mathbf{n}\right|}\right)\to 0,\end{array}$$

(1)

$$\begin{array}{c}\hfill \mathbb{P}\left(\left|\sum _{\mathbf{i}⪯\mathbf{n}}(Y_{\mathbf{n},\mathbf{i}}-\mathbb{E}\left[Y_{\mathbf{n},\mathbf{i}}\right])\right|>\epsilon b_{\left|\mathbf{n}\right|}\right)\to 0,\end{array}$$

(2)

and

$$\begin{array}{c}\hfill b_{\left|\mathbf{n}\right|}^{-1}\left|\sum _{\mathbf{i}⪯\mathbf{n}}\left(\mathbb{E}\left[Y_{\mathbf{n},\mathbf{i}}\right]-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}\right]\right)\right|\to 0.\end{array}$$

(3)

It is straightforward to confirm (1) via $\left(i\right)$ that

$$\begin{array}{ccc}& & \mathbb{P}\left(\left|\sum _{\mathbf{i}⪯\mathbf{n}}(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}-Y_{\mathbf{n},\mathbf{i}})\right|>\epsilon b_{\left|\mathbf{n}\right|}\right)\hfill \\ & \le & \mathbb{P}\left(\bigcup _{\mathbf{i}⪯\mathbf{n}}\{a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}\ne Y_{\mathbf{n},\mathbf{i}}\}\right)\le \sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b_{\left|\mathbf{n}\right|})\to 0.\hfill \end{array}$$

For (2), noting by Lemma 2 that $\{Y_{\mathbf{n},\mathbf{i}}-\mathbb{E}\left[Y_{\mathbf{n},\mathbf{i}}\right],\mathbf{1}⪯\mathbf{i}⪯\mathbf{n},\mathbf{n}\in {\mathbb{Z}}_{+}^d\}$ is still pairwise NQD, a combination of Markov inequality, Lemma 4, $\left(i\right)$ and $\left(ii\right)$ gives

$$\begin{array}{ccc}& & \mathbb{P}\left(\left|\sum _{\mathbf{i}⪯\mathbf{n}}(Y_{\mathbf{n},\mathbf{i}}-\mathbb{E}\left[Y_{\mathbf{n},\mathbf{i}}\right])\right|>\epsilon b_{\left|\mathbf{n}\right|}\right)\hfill \\ & \le & {\epsilon }^{-2}{b}_{\left|\mathbf{n}\right|}^{-2}\mathbb{E}\left({\left|\sum _{\mathbf{i}⪯\mathbf{n}}(Y_{\mathbf{n},\mathbf{i}}-\mathbb{E}\left[Y_{\mathbf{n},\mathbf{i}}\right])\right|}^2\right)\hfill \\ & \le & C{b}_{\left|\mathbf{n}\right|}^{-2}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[\right|Y_{\mathbf{n},\mathbf{i}}{|}^2]\hfill \\ & =& C\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b_{\left|\mathbf{n}\right|})+C{b}_{\left|\mathbf{n}\right|}^{-2}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^2{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}]\hfill \\ & \to & 0asn\to \infty .\hfill \end{array}$$

Finally, it follows from $\left(i\right)$ again that

$$\begin{array}{ccc}& & b_{\left|\mathbf{n}\right|}^{-1}\left|\sum _{\mathbf{i}⪯\mathbf{n}}\left(\mathbb{E}\left[Y_{\mathbf{n},\mathbf{i}}\right]-\mathbb{E}\left[a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}\right]\right)\right|\hfill \\ & \le & b_{\left|\mathbf{n}\right|}^{-1}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[\right|Y_{\mathbf{n},\mathbf{i}}-a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{\left|\mathbf{n}\right|}\}}\left|\right]\hfill \\ & =& \sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b_{\left|\mathbf{n}\right|})\to 0.\hfill \end{array}$$

This establishes (3), thereby concluding the proof. □

Proof of Corollary 1.

To derive Corollary 1, it is enough to check that conditions $\left(i\right)$ and $\left(ii\right)$ of Theorem 3 are satisfied with $b_{\left|\mathbf{n}\right|}=\left|\mathbf{n}\right|$. Actually, it follows from $a{\int }_0^b{x}^{a-1}\mathbb{P}\left(\right|\zeta |>x)dx=b^a\mathbb{P}\left(\right|\zeta |>b)+{\mathbb{E}\left[\right|\zeta |}^a{1}_{\left\{\right|\zeta |\le b\}}]$ and $k\mathbb{P}\left(\right|X|>k)\to 0$ as $k\to \infty$ that for all $\epsilon >0$, there is $K=K\left(\epsilon \right)$ such that

$$\begin{array}{ccc}& & \sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>b_{|\mathbf{n}|})\le M\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>|\mathbf{n}|)\hfill \\ & =& M\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>|\mathbf{n}|,|X|>|\mathbf{n}|)+M\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>|\mathbf{n}|,|X|\le |\mathbf{n}|)\hfill \\ & \le & M\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|X|>|\mathbf{n}|)+M{|\mathbf{n}|}^{-q}\sum _{\mathbf{i}⪯\mathbf{n}}|a_{\mathbf{n},\mathbf{i}}{|}^q{\mathbb{E}[|X|}^q{1}_{\{|X|\le |\mathbf{n}|\}}]\hfill \\ & \le & C{|\mathbf{n}|}^{1-q}\left[{|\mathbf{n}|}^q\mathbb{P}(|X|>|\mathbf{n}|)+{\mathbb{E}[|X|}^q{1}_{\{|X|\le |\mathbf{n}|\}}]\right]\hfill \\ & =& Cq{|\mathbf{n}|}^{1-q}{\int }_0^{|\mathbf{n}|}{x}^{q-1}\mathbb{P}(|X|>x)dx\hfill \\ & =& Cq{|\mathbf{n}|}^{1-q}{\int }_0^1{x}^{q-1}\mathbb{P}(|X|>x)dx+Cq{|\mathbf{n}|}^{1-q}{\displaystyle \sum _{k=1}^{|\mathbf{n}|-1}}{\int }_k^{k+1}{x}^{q-1}\mathbb{P}(|X|>x)dx\hfill \\ & \le & o\left(1\right)+C{|\mathbf{n}|}^{1-q}{\displaystyle \sum _{k=1}^{|\mathbf{n}|-1}}{k}^{q-1}\mathbb{P}(|X|>k)\hfill \\ & \le & o\left(1\right)+C{|\mathbf{n}|}^{1-q}{\displaystyle \sum _{k=1}^K}{k}^{q-1}\mathbb{P}(|X|>k)+C\epsilon {|\mathbf{n}|}^{1-q}{\displaystyle \sum _{k=K}^{|\mathbf{n}|-1}}{k}^{q-2}\hfill \\ & \le & o\left(1\right)+C\epsilon .\hfill \end{array}$$

As $\epsilon >0$ is arbitrary, this verifies condition $\left(i\right)$ immediately. Similarly, for condition $\left(ii\right)$, by virtue of Lemma 5 and the procedure above, we also have

$$\begin{array}{ccc}& & b_{|\mathbf{n}|}^{-2}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^2{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le b_{|\mathbf{n}|}\}}]\hfill \\ & \le & M{|\mathbf{n}|}^{-2}\sum _{\mathbf{i}⪯\mathbf{n}}\left[{|\mathbf{n}|}^2\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>|\mathbf{n}|)+\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^2{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le |\mathbf{n}|\}}]\right]\hfill \\ & =& o\left(1\right)+M{|\mathbf{n}|}^{-2}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^2{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le |\mathbf{n}|\}}]\hfill \\ & =& o\left(1\right)+M{|\mathbf{n}|}^{-2}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^2{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le |\mathbf{n}|,|X|>|\mathbf{n}|\}}]\hfill \\ & & +M{|\mathbf{n}|}^{-2}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^2{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le |\mathbf{n}|,|X|\le |\mathbf{n}|\}}]\hfill \\ & \le & o\left(1\right)+M\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|X|>|\mathbf{n}|)+M{|\mathbf{n}|}^{-q}\sum _{\mathbf{i}⪯\mathbf{n}}|a_{\mathbf{n},\mathbf{i}}{|}^q{\mathbb{E}[|X|}^q{1}_{\{|X|\le |\mathbf{n}|\}}]\hfill \\ & \to & 0\mathrm{as}|\mathbf{n}|\to \infty .\hfill \end{array}$$

Therefore, the proof of Corollary 1 is completed. □

Proof of Theorem 4.

It may also be assumed without loss of generality that $a_{\mathbf{n},\mathbf{i}}\ge 0$ for each $\mathbf{1}⪯\mathbf{i}⪯\mathbf{n}$, $\mathbf{n}\in {\mathbb{Z}}_{+}^d$. For given $\mathbf{1}⪯\mathbf{i}⪯\mathbf{n}$, we define

$$\begin{array}{ccc}\hfill U_{\mathbf{n},\mathbf{i}}& =& -{\Delta }_{\left|\mathbf{n}\right|}I(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}<-{\Delta }_{\left|\mathbf{n}\right|})+a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}I\left(\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {\Delta }_{\left|\mathbf{n}\right|})+{\Delta }_{\left|\mathbf{n}\right|}I(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}>{\Delta }_{\left|\mathbf{n}\right|}),\hfill \\ \hfill V_{\mathbf{n},\mathbf{i}}& =& a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}-U_{\mathbf{n},\mathbf{i}}=(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}+{\Delta }_{\left|\mathbf{n}\right|})I(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}<-{\Delta }_{\left|\mathbf{n}\right|})+(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}-{\Delta }_{\left|\mathbf{n}\right|})I(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}>{\Delta }_{\left|\mathbf{n}\right|}).\hfill \end{array}$$

It follows from Lemma 2 that the zero-mean property and pairwise NQD dependence are preserved for both $\{U_{\mathbf{n},\mathbf{i}}-\mathbb{E}[U_{\mathbf{n},\mathbf{i}}]\}$ and $\{V_{\mathbf{n},\mathbf{i}}-\mathbb{E}[V_{\mathbf{n},\mathbf{i}}]\}$ over $\mathbf{1}⪯\mathbf{i}⪯\mathbf{n}$, $\mathbf{n}\in {\mathbb{Z}}_{+}^d$. Using Hölder’s inequality ${\sum }_{\mathbf{i}⪯\mathbf{n}}|{\alpha }_{\mathbf{n},\mathbf{i}}{\beta }_{\mathbf{n},\mathbf{i}}|\le ({\sum }_{\mathbf{i}⪯\mathbf{n}}|{\alpha }_{\mathbf{n},\mathbf{i}}{|}^r{)}^{1/r}({\sum }_{\mathbf{i}⪯\mathbf{n}}|{\beta }_{\mathbf{n},\mathbf{i}}{{|}^s)}^{1/s}$, $r^{-1}+s^{-1}=1$ with ${\alpha }_{\mathbf{n},\mathbf{i}}=a_{\mathbf{n},\mathbf{i}}^{q^{\prime }}$, ${\beta }_{\mathbf{n},\mathbf{i}}=1$, and $r=q/q^{\prime }$, we have that for any $q^{\prime }\in (0,q)$,

$$\begin{array}{c}\hfill \sum _{\mathbf{i}⪯\mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}^{q^{\prime }}\le {\left(\sum _{\mathbf{i}⪯\mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}^q\right)}^{q^{\prime }/q}{\left(\sum _{\mathbf{i}⪯\mathbf{n}}1\right)}^{1-q^{\prime }/q}=O\left(\left|\mathbf{n}\right|\right).\end{array}$$

(4)

We first consider the case $1<p<2$. Choose $s=\min \{2,q\}$ and $t=\max \{1,r\}$. On one hand, it follows from Lemma 4, $C_r$-inequality, Jensen’s inequality, Lemma 5, and (4) that

$$\begin{array}{ccc}& & \mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(U_{\mathbf{n},\mathbf{i}}-\mathbb{E}[U_{\mathbf{n},\mathbf{i}}])\right|\right)}^s\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|U_{\mathbf{n},\mathbf{i}}-\mathbb{E}[U_{\mathbf{n},\mathbf{i}}]{|}^s]\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|U_{\mathbf{n},\mathbf{i}}{|}^s]\hfill \\ & =& C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^s{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {\Delta }_{|\mathbf{n}|}\}}]+C\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{\Delta }_{|\mathbf{n}|})\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X|}^s{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le {\Delta }_{|\mathbf{n}|}\}}]+C\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>{\Delta }_{|\mathbf{n}|})\hfill \\ & =& C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X|}^s{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le {\Delta }_{|\mathbf{n}|},|X|\le {\Delta }_{|\mathbf{n}|}\}}]+C\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>{\Delta }_{|\mathbf{n}|},|X|\le {\Delta }_{|\mathbf{n}|})\hfill \\ & & +C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X|}^s{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le {\Delta }_{|\mathbf{n}|},|X|>{\Delta }_{|\mathbf{n}|}\}}]+C\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>{\Delta }_{|\mathbf{n}|},|X|>{\Delta }_{|\mathbf{n}|})\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X|}^s{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le {\Delta }_{|\mathbf{n}|},|X|\le {\Delta }_{|\mathbf{n}|}\}}]+C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^s{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|>{\Delta }_{|\mathbf{n}|},|X|\le {\Delta }_{|\mathbf{n}|}\}}]\hfill \\ & & +C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X|}^s{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|\le {\Delta }_{|\mathbf{n}|},|X|>{\Delta }_{|\mathbf{n}|}\}}]+C\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|a_{\mathbf{n},\mathbf{i}}X|>{\Delta }_{|\mathbf{n}|},|X|>{\Delta }_{|\mathbf{n}|})\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}|a_{\mathbf{n},\mathbf{i}}{|}^s{\mathbb{E}[|X|}^s{1}_{\{|X|\le {\Delta }_{|\mathbf{n}|}\}}]+C\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}(|X|>{\Delta }_{|\mathbf{n}|})\hfill \\ & =& C|\mathbf{n}|{\Delta }_{|\mathbf{n}|}^{-s}{\mathbb{E}[|X|}^s{1}_{\{|X|\le {\Delta }_{|\mathbf{n}|}\}}]+C|\mathbf{n}|\mathbb{P}(|X|>{\Delta }_{|\mathbf{n}|}).\hfill \end{array}$$

(5)

The convergence to zero of the second term in the last line of (5) follows from the hypothesis that $n\mathbb{P}\left(\right|X|>{\Delta }_n)\to 0$ when $n\to \infty$. Thus, in what follows, we will deal with the first term. Actually, it is evident that

$$\begin{array}{ccc}\hfill \left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{\mathbb{E}\left[\right|X|}^s{1}_{\left\{\right|X|\le {\Delta }_{\left|\mathbf{n}\right|}\}}]& \le & \left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{\int }_0^{{\Delta }_{\left|\mathbf{n}\right|}^s}\mathbb{P}\left(\right|X|>x^{1/s})dx\hfill \\ & \le & \left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{\Delta }_1^s+\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{\displaystyle \sum _{k=1}^{\left|\mathbf{n}\right|-1}}{\int }_{{\Delta }_k^s}^{{\Delta }_{k+1}^s}\mathbb{P}\left(\right|X|>x^{1/s})dx\hfill \\ & \le & o(1)+{\displaystyle \sum _{k=1}^{\left|\mathbf{n}\right|-1}}\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{\displaystyle \frac{{\Delta }_{k+1}^s-{\Delta }_k^s}{k}}·k\mathbb{P}\left(\right|X|>{\Delta }_k).\hfill \end{array}$$

(6)

It follows from Lemma 6 that

$$\begin{array}{c}\hfill {\displaystyle \sum _{k=1}^{\left|\mathbf{n}\right|-1}}\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{\displaystyle \frac{{\Delta }_{k+1}^s-{\Delta }_k^s}{k}}\le C{\displaystyle \sum _{k=1}^{\left|\mathbf{n}\right|-1}}\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{k}^{s/p-2}{\mathcal{l}}^s\left(k\right)\le C,\end{array}$$

and for each fixed $1\le k\le \left|\mathbf{n}\right|$,

$$\begin{array}{c}\hfill \left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{\displaystyle \frac{{\Delta }_{k+1}^s-{\Delta }_k^s}{k}}\le C\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{k}^{s/p-2}{\mathcal{l}}^s\left(k\right)\to 0\mathrm{as}\left|\mathbf{n}\right|\to \infty .\end{array}$$

Hence, we obtain by the Toeplitz lemma and the assumption $n\mathbb{P}\left(\right|X|>{\Delta }_n)\to 0$ as $n\to \infty$ that

$$\begin{array}{c}\hfill {\displaystyle \sum _{k=1}^{\left|\mathbf{n}\right|-1}}\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{\displaystyle \frac{{\Delta }_{k+1}^s-{\Delta }_k^s}{k}}·k\mathbb{P}\left(\right|X|>{\Delta }_k)\to 0\mathrm{as}\left|\mathbf{n}\right|\to \infty ,\end{array}$$

(7)

which together with (6) derives that the first term in the last line of (5) converges to zero.

On the other hand, applying Lemma 4, the $C_r$-inequality, Jensen’s inequality, Lemma 5, and (4) once more yields

$$\begin{array}{ccc}& & \mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(V_{\mathbf{n},\mathbf{i}}-\mathbb{E}[V_{\mathbf{n},\mathbf{i}}])\right|\right)}^t\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-t}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[|V_{\mathbf{n},\mathbf{i}}-\mathbb{E}[V_{\mathbf{n},\mathbf{i}}]{|}^t\right]\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-t}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|V_{\mathbf{n},\mathbf{i}}{|}^t]\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-t}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^t{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{\Delta }_{|\mathbf{n}|}\}}]\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-t}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^t{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|>{\Delta }_{|\mathbf{n}|}\}}]\hfill \\ & =& C{\Delta }_{|\mathbf{n}|}^{-t}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^t{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|>{\Delta }_{|\mathbf{n}|},|X|\le {\Delta }_{|\mathbf{n}|}\}}]\hfill \\ & & +C{\Delta }_{|\mathbf{n}|}^{-t}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^t{1}_{\{|a_{\mathbf{n},\mathbf{i}}X|>{\Delta }_{|\mathbf{n}|},|X|>{\Delta }_{|\mathbf{n}|}\}}]\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-s}\sum _{\mathbf{i}⪯\mathbf{n}}|a_{\mathbf{n},\mathbf{i}}{|}^s{\mathbb{E}[|X|}^s{1}_{\{|X|\le {\Delta }_{|\mathbf{n}|}\}}]+C{\Delta }_{|\mathbf{n}|}^{-t}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}X{|}^t{1}_{\{|X|>{\Delta }_{|\mathbf{n}|}\}}]\hfill \\ & =& C|\mathbf{n}|{\Delta }_{|\mathbf{n}|}^{-s}{\mathbb{E}[|X|}^s{1}_{\{|X|\le {\Delta }_{|\mathbf{n}|}\}}]+C|\mathbf{n}|{\Delta }_{|\mathbf{n}|}^{-t}{\mathbb{E}[|X|}^t{1}_{\{|X|>{\Delta }_{|\mathbf{n}|}\}}].\hfill \end{array}$$

(8)

Thanks to (6) and (7), the first term in the last line of (8) is seen to converge to zero. Therefore, the rest of the proof focuses on the second term. The convergence $n\mathbb{P}(|X|>{\Delta }_n)\to 0$ as $n\to \infty$ implies that, for any $\epsilon >0$, we can choose $N\left(\epsilon \right)$ (depending only on $\epsilon$) so that $n\mathbb{P}\left(\right|X|>{\Delta }_n)<\epsilon$ holds for each $n>N\left(\epsilon \right)$. By the fact that ${\mathbb{E}\left[\right|\zeta |}^a{1}_{\left\{\right|\zeta |>b\}}]=b^a\mathbb{P}\left(\right|\zeta |>b)+{\int }_{b^a}^{\infty }\mathbb{P}\left(\right|\zeta |>x^{1/a})dx$ and Lemma 6, we have that for all $\left|\mathbf{n}\right|>N\left(\epsilon \right)$,

$$\begin{array}{ccc}& & \left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-t}{\mathbb{E}\left[\right|X|}^t{1}_{\left\{\right|X|>{\Delta }_{\left|\mathbf{n}\right|}\}}]\hfill \\ & =& \left|\mathbf{n}\right|\mathbb{P}\left(\right|X|>{\Delta }_{\left|\mathbf{n}\right|})+\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-t}{\int }_{{\Delta }_{\left|\mathbf{n}\right|}^t}^{\infty }\mathbb{P}\left(\right|X|>x^{1/t})dx\hfill \\ & =& o\left(1\right)+\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-t}{\displaystyle \sum _{k=\left|\mathbf{n}\right|}^{\infty }}{\int }_{{\Delta }_k^t}^{{\Delta }_{k+1}^t}\mathbb{P}\left(\right|X|>x^{1/t})dx\hfill \\ & \le & o\left(1\right)+\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-t}{\displaystyle \sum _{k=\left|\mathbf{n}\right|}^{\infty }}{\displaystyle \frac{{\Delta }_{k+1}^t-{\Delta }_k^t}{k}}·k\mathbb{P}\left(\right|X|>{\Delta }_k)\hfill \\ & \le & o\left(1\right)+C\epsilon \left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-t}{\displaystyle \sum _{k=\left|\mathbf{n}\right|}^{\infty }}{k}^{t/p-2}{\mathcal{l}}^t\left(k\right)\hfill \\ & \le & o\left(1\right)+C\epsilon ,\hfill \end{array}$$

which, together with the arbitrariness of $\epsilon >0$, verifies that the second term of (8) converges to zero.

Consequently, in view of the arguments above, we apply $C_r$-inequality and Jensen’s inequality to obtain

$$\begin{array}{ccc}\hfill \mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}{a}_{\mathbf{n},\mathbf{i}}(X_{\mathbf{i}}-\mathbb{E}[X_{\mathbf{i}}])\right|\right)}^r& \le & 2^{r-1}\mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(U_{\mathbf{n},\mathbf{i}}-\mathbb{E}[U_{\mathbf{n},\mathbf{i}}])\right|\right)}^r\hfill \\ & & +2^{r-1}\mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(V_{\mathbf{n},\mathbf{i}}-\mathbb{E}[V_{\mathbf{n},\mathbf{i}}])\right|\right)}^r\hfill \\ & \le & 2^{r-1}{\left[\mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(U_{\mathbf{n},\mathbf{i}}-\mathbb{E}[U_{\mathbf{n},\mathbf{i}}])\right|\right)}^s\right]}^{r/s}\hfill \\ & & +2^{r-1}{\left[\mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(V_{\mathbf{n},\mathbf{i}}-\mathbb{E}[V_{\mathbf{n},\mathbf{i}}])\right|\right)}^t\right]}^{r/t}\hfill \\ & \to & 0as\left|\mathbf{n}\right|\to \infty .\hfill \end{array}$$

The argument for the case $1<p<2$ is thereby completed.

For $p=1$, note that

$$\begin{array}{ccc}& & \mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}\left(a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}-\mathbb{E}[a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {\Delta }_{\left|\mathbf{n}\right|}\}}]\right)\right|\right)}^r\hfill \\ & \le & \mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(U_{\mathbf{n},\mathbf{i}}-\mathbb{E}[U_{\mathbf{n},\mathbf{i}}])\right|\right)}^r+\mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}{V}_{\mathbf{n},\mathbf{i}}\right|\right)}^r\hfill \\ & & +{\left(\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{\Delta }_{\left|\mathbf{n}\right|})\right)}^r\hfill \\ & \le & \mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(U_{\mathbf{n},\mathbf{i}}-\mathbb{E}[U_{\mathbf{n},\mathbf{i}}])\right|\right)}^r+{\Delta }_{\left|\mathbf{n}\right|}^{-r}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[\right|V_{\mathbf{n},\mathbf{i}}{|}^r]\hfill \\ & & +{\left(\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{\Delta }_{\left|\mathbf{n}\right|})\right)}^r.\hfill \end{array}$$

It follows from (5)–(7) that as $\left|\mathbf{n}\right|\to \infty$,

$$\begin{array}{c}\hfill \mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(U_{\mathbf{n},\mathbf{i}}-\mathbb{E}[U_{\mathbf{n},\mathbf{i}}])\right|\right)}^r\le {\left[\mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\left|\sum _{\mathbf{i}⪯\mathbf{n}}(U_{\mathbf{n},\mathbf{i}}-\mathbb{E}[U_{\mathbf{n},\mathbf{i}}])\right|\right)}^s\right]}^{r/s}\to 0\end{array}$$

and

$$\begin{array}{c}\hfill {\left(\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{P}\left(\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{\Delta }_{\left|\mathbf{n}\right|})\right)}^r\le C{\left(\left|\mathbf{n}\right|{\Delta }_{\left|\mathbf{n}\right|}^{-s}{\mathbb{E}\left[\right|X|}^s{1}_{\left\{\right|X|\le {\Delta }_{\left|\mathbf{n}\right|}\}}]+C\left|\mathbf{n}\right|\mathbb{P}(\left|X\right|>{\Delta }_{\left|\mathbf{n}\right|})\right)}^r\to 0.\end{array}$$

Moreover, noting that $t=\max \{1,r\}$, by (8), one can also verify that

$$\begin{array}{ccc}& & {\Delta }_{|\mathbf{n}|}^{-r}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|V_{\mathbf{n},\mathbf{i}}{|}^r]\hfill \\ & =& C{\Delta }_{|\mathbf{n}|}^{-r}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^r{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{\Delta }_{|\mathbf{n}|}\}}]\hfill \\ & \le & C{\Delta }_{|\mathbf{n}|}^{-t}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}[|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^t{1}_{\{|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{\Delta }_{|\mathbf{n}|}\}}]\hfill \\ & \le & C|\mathbf{n}|{\Delta }_{|\mathbf{n}|}^{-s}{\mathbb{E}[|X|}^s{1}_{\{|X|\le {\Delta }_{|\mathbf{n}|}\}}])+C|\mathbf{n}|{\Delta }_{|\mathbf{n}|}^{-t}{\mathbb{E}[|X|}^t{1}_{\{|X|>{\Delta }_{|\mathbf{n}|}\}}]\to 0.\hfill \end{array}$$

□

Proof of Theorem 5.

Let $\tau =\min \{q,1\}$. Note that $0<r<p<1$ and $q>p$, so $\tau \in (p,1]$ and $r/\tau <1$. We begin with the decomposition

$$\begin{array}{cc}\hfill & \mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\underset{\mathbf{k}⪯\mathbf{n}}{\max }\left|\sum _{\mathbf{i}⪯\mathbf{k}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}\right|\right)}^r\hfill \\ \hfill & \le \mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\underset{\mathbf{k}⪯\mathbf{n}}{\max }\left|\sum _{\mathbf{i}⪯\mathbf{k}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {\Delta }_{\left|\mathbf{n}\right|}\}}\right|\right)}^r\hfill \\ \hfill & +\mathbb{E}{\left({\displaystyle \frac{1}{{\Delta }_{\left|\mathbf{n}\right|}}}\underset{\mathbf{k}⪯\mathbf{n}}{\max }\left|\sum _{\mathbf{i}⪯\mathbf{k}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{\Delta }_{\left|\mathbf{n}\right|}\}}\right|\right)}^r\hfill \\ \hfill & =:I_1+I_2.\hfill \end{array}$$

For $I_1$, we apply the inequality $|{\sum }_j{y}_j{|}^{\tau }\le {\sum }_j{\left|y_j\right|}^{\tau }$ for $0<\tau \le 1$ to the inner sum, then use the fact that the maximum over $\mathbf{k}⪯\mathbf{n}$ is bounded by the sum over the full index set $\mathbf{i}⪯\mathbf{n}$. This yields

$$\begin{array}{cc}\hfill \underset{\mathbf{k}⪯\mathbf{n}}{\max }{\left|\sum _{\mathbf{i}⪯\mathbf{k}}{a}_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {\Delta }_{\left|\mathbf{n}\right|}\}}\right|}^{\tau }& \le \underset{\mathbf{k}⪯\mathbf{n}}{\max }\sum _{\mathbf{i}⪯\mathbf{k}}{\left|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}\right|}^{\tau }{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {\Delta }_{\left|\mathbf{n}\right|}\}}\hfill \\ \hfill & \le \sum _{\mathbf{i}⪯\mathbf{n}}{\left|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}\right|}^{\tau }{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {\Delta }_{\left|\mathbf{n}\right|}\}},\hfill \end{array}$$

which together with Jensen’s inequality yields

$$I_1\le {\left[{\Delta }_{\left|\mathbf{n}\right|}^{-\tau }\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^{\tau }{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|\le {\Delta }_{\left|\mathbf{n}\right|}\}}]\right]}^{r/\tau }$$

Applying inequality $|{\sum }_j{y}_j{|}^r\le {\sum }_j{\left|y_j\right|}^r$ to $I_2$ gives

$$I_2\le {\Delta }_{\left|\mathbf{n}\right|}^{-r}\sum _{\mathbf{i}⪯\mathbf{n}}\mathbb{E}\left[\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}{|}^r{1}_{\left\{\right|a_{\mathbf{n},\mathbf{i}}{X}_{\mathbf{i}}|>{\Delta }_{\left|\mathbf{n}\right|}\}}].$$

Both bounds now involve only sums over the full index set $\mathbf{i}⪯\mathbf{n}$, with the dependence on the partial index $\mathbf{k}$ completely removed. The rest of the argument coincides precisely with that of Theorem 4: apply Lemma 5 to replace $X_{\mathbf{i}}$ by X, use the moment condition ${\sum }_{\mathbf{i}⪯\mathbf{n}}|a_{\mathbf{n},\mathbf{i}}{|}^q=O\left(\right|\mathbf{n}\left|\right)$ to control the weight sums, and employ the slicing argument together with the Toeplitz lemma and the assumption $n\mathbb{P}\left(\right|X|>{\Delta }_n)\to 0$ to show that both $I_1$ and $I_2$ converge to zero as $\left|\mathbf{n}\right|\to \infty$. □