Strong laws for weighted sums of widely orthant dependent random variables and applications

Yong Zhu; Wei Wang; Kan Chen

doi:10.1515/math-2024-0027

Article Open Access

Strong laws for weighted sums of widely orthant dependent random variables and applications

Yong Zhu , Wei Wang and Kan Chen

Published/Copyright: August 26, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this study, the strong law of large numbers and the convergence rate for weighted sums of non-identically distributed widely orthant dependent random variables are established. As applications, the strong consistency for weighted estimator in nonparametric regression model and the rate of strong consistency for least-squares estimator in multiple linear regression model are obtained. Some numerical simulations are also provided to verify the validity of the theoretical results.

Keywords: strong law of large numbers; weighted sums; widely orthant dependent; strong consistency

MSC 2010: 60F15; 62G05

1 Introduction

It is known that the strong laws of large numbers (SLLN) are fundamental theorems in probability theory. Moreover, many well-known estimators in statistics, such as the least squares (LS) estimators, nonparametric regression function estimators, and jackknife estimators, are the forms of weighted sums of random variables. Therefore, it is desirable and of great interest to study the limit behaviors for the weighted sums of random variables. The central limit theorem and other weak laws for the sums of independent and identically distributed (i.i.d.) random variables have been extended to the weighted sums and many other dependence cases in the past few decades. However, the study on SLLN for the weighted sums of dependent random variables is still a hop topic in recent years.

This study will mainly consider the SLLN for weighted sums of a quite general dependent random variables, i.e., widely orthant dependent (WOD) random variables. The concept of WOD random variables was put forward by Wang et al. [1] as follows.

Definition 1.1

Random variables X 1 , X 2 , … , X n are said to be widely upper orthant dependent (WUOD) if there is a finite positive number g U ( n ) such that for all x i ∈ R , 1 ≤ i ≤ n ,

P ( X 1 > x 1 , X 2 > x 2 , … , X n > x n ) ≤ g U ( n ) ∏ i = 1 n P ( X i > x i ) .

Random variables X 1 , X 2 , … , X n are said to be widely lower orthant dependent (WLOD) if there is a finite positive number g L ( n ) such that for all x i ∈ R , 1 ≤ i ≤ n ,

P ( X 1 ≤ x 1 , X 2 ≤ x 2 , … , X n ≤ x n ) ≤ g L ( n ) ∏ i = 1 n P ( X i ≤ x i ) .

If X 1 , X 2 , … , X n are WUOD as well as WLOD, they are called WOD, where g U ( n ) , g L ( n ) are dominating coefficients. A sequence { X n , n ≥ 1 } of random variables is said to be WOD if every finite sub-collection is WOD. An array { X n i , 1 ≤ i ≤ n , n ≥ 1 } of random variables is said to be rowwise WOD if for every n ≥ 1 , the sequence { X n i , 1 ≤ i ≤ n } is WOD.

By letting x i → − ∞ and x i → ∞ , 1 ≤ i ≤ n , respectively, we have g U ( n ) ≥ 1 and g L ( n ) ≥ 1 . The random sequence is said to be extended negatively dependent (END), the concept of which was proposed by Liu [2], if g U ( n ) = g L ( n ) = M ≥ 1 for all n ≥ 1 . Furthermore, if M = 1 , then the sequence is negatively orthant dependent (NOD). The concept of NOD random variables was first introduced by Lehmann [3], and then carefully studied by Joag-Dev and Proschan [4] who stated that NA random variables are NOD but not vice versa. Furthermore, Hu [5] pointed out that negatively supper-additive dependent (NSD) random variables include NA random variables but are NOD. Hence, the WOD random sequences include independent sequences, NA sequences, NSD sequences, NOD sequences, and END sequences as special cases. Wang et al. [1] presented some examples to show that WOD random variables contain some negatively dependent random variables, positively dependent random variables, and some other classes of dependent random variables. Thus, the study of WOD random variables is of great interest. There have been a number of works on the WOD random variables. Some corresponding studies include Wang and Cheng [6], Chen et al. [7], Shen [8], Wang et al. [9], Yang et al. [10], Wang and Hu [11], Naderi et al. [12], and Xia et al. [13].

In this work, some further results on the SLLN for weighted sums of WOD random variables are established. The random variables are not required to be identically distributed or stochastically dominated by another random variable. Moreover, the restriction on the dominating coefficients is rather weak, and they can increase polynomially even geometrically. As a direct corollary, the Kolmogorov SLLN for non-identically distributed WOD random variables is obtained under very general conditions. As applications of the main results, the strong consistency for weighted estimator in nonparametric regression model and the rate of strong consistency for LS estimator in multiple linear regression model are obtained. Some simulation studies are also carried out to support these theorems.

The outline of this work is stated as follows. The main results on SLLN and applications to nonparametric regression model as well as multiple linear regression model are given in Section 2. Numerical simulations are presented in Section 3. The proofs of the results are stated in Section 4. Throughout this article, C , C i , i ≥ 1 represent the positive constants whose values can be different in different places. Let log x = ln max { x , e } , and I ( A ) be the indicator function of the set A . ⌊ x ⌋ stands for the integer part of x not exceeding x . c n = o ( b n ) implies c n ⁄ b n → 0 , and c n = O ( b n ) means c n ≤ C b n for all n .

2 Main results and applications

2.1 Main results

Now, we present our main results. The first one is the strong law of large numbers for weighted sums of non-identically distributed WOD random variables.

Theorem 2.1

Assume that { X n , n ≥ 1 } is a sequence of zero mean WOD random variables and { a n i , 1 ≤ i ≤ n , n ≥ 1 } is an array of real numbers. Suppose that one of the following assumptions holds:

The dominating coefficients g ( n ) = O ( n τ ) for some τ ≥ 0 , max 1 ≤ i ≤ n ∣ a n i ∣ = O ( n − 1 ⁄ α ) for some α ≥ 2 and ∑ i = 1 n a n i 2 = o ( log − 1 n ) ;
The dominating coefficients g ( n ) = O ( e n ϱ ) for some 0 ≤ ϱ < ( p − α ) ⁄ ( α p − α ) with p > α ≥ 1 , max 1 ≤ i ≤ n ∣ a n i ∣ = O ( n − 1 ⁄ α ) , and ∑ i = 1 n ∣ a n i ∣ = O ( 1 ) .

If sup i ≥ 1 E ∣ X i ∣ p < ∞ for some p > α , then

∑ i = 1 n a n i X i → 0 , a.s. , n → ∞ .

Remark 2.1

It is known that the Marcinkiewicz-Zygmund-type SLLN n − 1 ⁄ α ∑ i = 1 n X i → 0 a.s. , n → ∞ holds for a sequence { X n , n ≥ 1 } of i.i.d. random variables with zero means and E ∣ X 1 ∣ α < ∞ for some 1 ≤ α < 2 . Recently, Wu and Wang [14] obtained the SLLN for weighted sums of identically distributed WOD random variables. A special case of their result shows that ∑ i = 1 n a n i X i → 0 a.s. , n → ∞ under the assumptions ∑ i = 1 n ∣ a n i ∣ q = O ( n 1 − q ⁄ α ) and E ∣ X 1 ∣ α q ⁄ ( q − α ) < ∞ for some q > α with 0 < α < 2 . Noting that the random variables in our theorem are assumed to be non-identically distributed, the moment condition in Theorem 2.1 is not strong. Moreover, the first condition allows the dominating coefficients increase polynomially, while the second condition allows them increase geometrically, and for both two cases, the moment conditions are unrelated to the dominating coefficients.

Remark 2.2

Qiu and Chen [15] obtained the complete convergence for weighted sums by assuming that { X n , n ≥ 1 } is a sequence of WOD random variables stochastically dominated by a random variable X with E ∣ X ∣ 2 α < ∞ , max 1 ≤ i ≤ n ∣ a n i ∣ = O ( n − 1 ⁄ α ) for some α ≥ 1 , and ∑ n = 1 ∞ g ( n ) exp ( − u ⁄ ∑ i = 1 n a n i 2 ) < ∞ for any u > 0 . Lita da Silva [16] also established the similar result by assuming 0 < α < 2 . Although the a.s. convergence is weaker than complete convergence, the moment condition here is also much weaker.

Taking a n i ≡ 1 ⁄ n and α = 1 in Theorem 2.1, we obtain the Kolmogorov SLLN for non-identically distributed WOD random variables as follows.

Corollary 2.1

Assume that { X n , n ≥ 1 } is a sequence of zero mean WOD random variables with the dominating coefficients g ( n ) = O ( e n ϱ ) for some 0 ≤ ϱ < 1 . If sup i ≥ 1 E ∣ X i ∣ p < ∞ for some p > 1 , then

n − 1 ∑ i = 1 n X i → 0 , a.s. , n → ∞ .

Remark 2.3

Chen et al. [17] obtained the Kolmogorov SLLN for identically distributed WOD random variables with the dominating coefficients g ( n ) = O ( n τ ) for some τ > 0 . The corresponding moment condition is E ∣ X ∣ log ∣ X ∣ < ∞ , which, to our knowledge, is the best among current studies. By the method of Rosalsky and Thành [18], the moment condition sup i ≥ 1 E ∣ X i ∣ log 1 + r ∣ X i ∣ < ∞ for some r > 1 is required for non-identically distributed WOD random variables. However, Corollary 2.1 allows the dominating coefficients increase geometrically, while the moment condition is slightly stronger. It is an open problem whether it can be further weakened and we leave it for future study.

The following result concerns the rate of strong convergence for weighted sums of non-identically distributed WOD random variables under some mild conditions.

Theorem 2.2

Assume that { X n , n ≥ 1 } is a sequence of zero mean WOD random variables with dominating coefficients g ( n ) = O ( n τ ) for some τ ≥ 0 , and { a n i , 1 ≤ i ≤ n , n ≥ 1 } is an array of real numbers satisfying max 1 ≤ i ≤ n ∣ a n i ∣ = O ( n − 1 ⁄ α ) for some α ≥ 2 and ∑ i = 1 n a n i 2 = O ( n − 2 ⁄ λ log − 1 n ) for some λ > α . If sup i ≥ 1 E ∣ X i ∣ p < ∞ for some p > λ α ⁄ ( λ − α ) , then

∑ i = 1 n a n i X i = O ( n − 1 ⁄ λ ) a.s.

Remark 2.4

From the proof of Theorem 2.2, one can easily see that if ∑ i = 1 n a n i 2 = O ( n − 2 ⁄ λ log − 1 n ) is replaced by ∑ i = 1 n a n i 2 = o ( n − 2 ⁄ λ log − 1 n ) , then ∣ ∑ i = 1 n a n i X i ∣ = o ( n − 1 ⁄ λ ) a.s.

2.2 Application to nonparametric regression model

In this subsection, we will investigate the nonparametric regression model as follows:

(2.1) Y n i = f ( x n i ) + ε n i , i = 1 , 2 , … , n , n ≥ 1 ,

where x n i are the known fixed design points from A , where A ⊂ R m is a given compact set for some m ≥ 1 , f ( ⋅ ) is an unknown regression function defined on A , and the ε n i are random errors. Assume that for each n ≥ 1 , the joint distribution of ( ε n 1 , ε n 2 , … , ε n n ) is the same as ( ε 1 , ε 2 , … , ε n ) . As an estimator of f ( ⋅ ) , we consider the following weighted regression estimator:

(2.2) f n ( x ) = ∑ i = 1 n ω n i ( x ) Y n i , x ∈ A ⊂ R m ,

where ω n i ( x ) = ω n i ( x ; x n 1 , x n 2 , … , x n n ) , i = 1 , 2 , … , n are weight functions.

The estimator was first proposed by Stone [19] and then applied to the fixed design case by Georgiev [20]. From then on, many scholars studied the limit properties of this estimator. For more details, we refer the reader to Roussas [21], Fan [22], Roussas et al. [23], Tran et al. [24], Liang and Jing [25], Wang et al. [9], Thanh and Yin [26], Shen [8], and Zhang et al. [27] among others.

Before presenting our result, we need the following notations. Let c ( f ) be the set of continuity points of the function f ( ⋅ ) on A and ‖ x ‖ be the Euclidean norm of x ∈ R m . For any x ∈ A , the following basic assumptions are indispensable:

( H 1 ) ∑ i = 1 n ω n i ( x ) → 1 , as n → ∞ ; ( H 2 ) max 1 ≤ i ≤ n ∣ ω n i ( x ) ∣ = O ( n − 1 ⁄ α ) , for some α ≥ 1 , ∑ i = 1 n ∣ ω n i ( x ) ∣ ≤ C < ∞ , for all n ; ( H 3 ) ∑ i = 1 n ∣ ω n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ > a ) → 0 as n → ∞ , for all a > 0 .

We can see that ( H 1 ) – ( H 3 ) are general assumptions, which are satisfied for the nearest neighbor weights. For more details, one can refer to Wang et al. [9]. With the assumptions in hand, we state the following result.

Theorem 2.3

Suppose that ( H 1 ) – ( H 3 ) hold. Let { ε n , n ≥ 1 } be a sequence of WOD random errors with dominating coefficients g ( n ) = O ( e n ϱ ) , for some 0 ≤ ϱ < ( p − α ) ⁄ ( α p − α ) with p > α . If sup i ≥ 1 E ∣ ε i ∣ p < ∞ , then for all x ∈ c ( f ) ,

(2.3) f n ( x ) → f ( x ) , a.s.

Remark 2.5

Liang and Jing [25] obtained the similar result for NA random variables, which is a special case of WOD random variables. Noting that WOD structure contains many negative dependence and positive dependence structures, therefore, our result extends that of Liang and Jing [25] to a much more general case.

Remark 2.6

Wang et al. [9] obtained the complete consistency for WOD random variables under the conditions that g ( n ) = O ( n κ ⁄ α ) for some 0 ≤ κ < 1 and { ε n , n ≥ 1 } is stochastically dominated by a random variable ε with E ∣ ε ∣ 2 α + κ < ∞ . Zhang et al. [27] improved the conditions to g ( n ) = O ( n ϑ ) for some ϑ ≥ 0 and E ∣ ε ∣ 2 α < ∞ . Although the strong consistency in Theorem 2.4 is slightly weaker than complete consistency, the assumption of stochastic domination is completely released here. Moreover, the moment condition is weaker and the dominating coefficients can increase geometrically.

2.3 Application to multiple linear regression model

In this subsection, we turn to study the following multiple linear regression model:

(2.4) y i = β 1 x i 1 + … + β p x i p + ε i ,

where x i j are the known constants, β = ( β 1 , … , β p ) T are the unknown parameters, Y n = ( y 1 , … , y n ) T are the observed values, and ε i , i ≥ 1 are the random errors. Let X n be the design matrix ( x i j ) 1 ≤ i ≤ n , 1 ≤ j ≤ p . Then, the LS estimator of β is given by

(2.5) β ˆ n = ( β ˆ n 1 , … , β ˆ n p ) T = ( X n T X n ) − 1 X n T Y n ,

provided that X n T X n is nonsingular. It can be seen in the study of Lai et al. [28] or Chen et al. [29] that

(2.6) β ˆ n j − β j = ∑ i = 1 n b n i ( j ) b n ( j ) ε i , 1 ≤ j ≤ p ,

where b n ( j ) = ∑ i = 1 n ( b n i ( j ) ) 2 = 1 v j j ( n ) , and ( v i j ( n ) ) p × p = ( X n T X n ) − 1 .

There are also some studies on the LS estimator (2.5) in the past few decades. Some works are listed as follows: Anderson and Taylor [30], Drygas [31], Lai et al. [28], Chen et al. [29], Baltagi and Krämer [32], Song [33], Hu et al. [34], Hu et al. [35], Lita da Silva and Mexia [36], Yang et al. [37], and references therein.

As an application of Theorem 2.2, the rate of strong consistency for (2.5) with WOD samples is stated as follows.

Theorem 2.4

Under Model (2.4), let { ε n , n ≥ 1 } be a sequence of zero mean WOD random errors with dominating coefficients g ( n ) = O ( n τ ) for some τ ≥ 0 . Suppose that max 1 ≤ i ≤ n ∣ b n i ( j ) ∣ ⁄ b n ( j ) = O ( n − 1 ⁄ α ) , liminf n → ∞ b n ( j ) ⁄ ( n 2 ⁄ λ log n ) ≥ c 0 for 1 ≤ j ≤ p , where λ > α ≥ 2 , c 0 is a positive constant. Then, sup n ≥ 1 E ∣ ε n ∣ p < ∞ for some p > λ α ⁄ ( λ − α ) implies

∣ β ˆ n − β ∣ = O ( n − 1 ⁄ λ ) a.s.

We present an example to show that the conditions of Theorem 2.4 are very general.

Example 2.1

For a simple linear regression model as

y i = β 1 + β 2 i + ε i , i ≥ 1 ,

Yang et al. [37] verified that

β ˆ n 1 − β 1 = 2 n ( n − 1 ) ∑ i = 1 n ( 2 n + 1 − 3 i ) ε i , β ˆ n 2 − β 2 = 6 n ( n 2 − 1 ) ∑ i = 1 n ( 2 i − n − 1 ) ε i ,

and b n ( 1 ) = n ( n − 1 ) 2 ( 2 n + 1 ) , b n ( 2 ) = n ( n 2 − 1 ) 12 . It is not difficult to verify that max 1 ≤ i ≤ n ∣ b n i ( j ) ∣ ⁄ b n ( j ) = O ( n − 1 ⁄ α ) , liminf n → ∞ b n ( j ) ⁄ ( n 2 ⁄ λ log n ) ≥ c 0 for j = 1 , 2 and any λ > α > 2 . Therefore, the rate of strong consistency can be easily obtained. For example, choosing α ≈ 2 , λ = 4 , then sup n ≥ 1 E ∣ ε n ∣ p < ∞ for some p > 4 implies ∣ β ˆ n − β ∣ = O ( n − 1 ⁄ 4 ) a.s.; if we choose α ≈ 2 , λ = 3 , then sup n ≥ 1 E ∣ ε n ∣ p < ∞ for some p > 6 implies ∣ β ˆ n − β ∣ = O ( n − 1 ⁄ 3 ) a.s.

3 Numerical simulation

In this section, some simple numerical simulations are conducted to study the finite sample performances of the estimators of f ( ⋅ ) and β in Section 2. We first generate the samples. Let ε ˜ i uniformly distributed on [ 0 , 1 ] for each 1 ≤ i ≤ n , while ( ε ˜ 1 , ε ˜ 2 ) , ( ε ˜ 3 , ε ˜ 4 ) , …, ( ε ˜ 2 m − 1 , ε ˜ 2 m ) , …, respectively, obey a joint distribution of the Farlie-Gumbel-Morgenstern (FGM) copula

(3.1) C θ n ( u , v ) = u v + θ n u v ( 1 − u ) ( 1 − v ) , ( u , v ) ∈ [ 0 , 1 ] × [ 0 , 1 ] .

Let ε n = ε ˜ n − E ε ˜ n . By Wang et al. [1] and Lemma 4.1, we have that { ε n , n ≥ 1 } is a sequence of zero mean WOD random variables. We first study the finite sample performance of the nearest neighbor weight function estimator under the assumption of Theorem 2.3.

Let A = [ 0 , 1 ] and x n i = i ⁄ n , i = 1 , 2 , … , n . For any x ∈ A , we reorder

∣ x n 1 − x ∣ , ∣ x n 2 − x ∣ , … , ∣ x n n − x ∣

as follows:

∣ x n , R 1 ( x ) − x ∣ ≤ ∣ x n , R 2 ( x ) − x ∣ ≤ … ≤ ∣ x n , R n ( x ) − x ∣ ,

if ∣ x n i − x ∣ = ∣ x n j − x ∣ , then ∣ x n i − x ∣ is permuted before ∣ x n j − x ∣ when x n i < x n j .

Let 1 ≤ k n ≤ n and define the nearest neighbor weight function as

ω ˜ n i ( x ) = 1 ⁄ k n , if ∣ x n i − x ∣ ≤ ∣ x n , R k n ( x ) − x ∣ , 0 , otherwise .

Choose θ n = n − 1 ⁄ 2 ⁄ 2 . It is easy to check by Wang et al. [1] that g ( n ) = O ( e n 1 ⁄ 2 ) . Take k n = ⌊ n 0.6 ⌋ . Choosing the points x = 0.01 , 0.02 , … , 0.99 , 1 and the sample sizes n = 200 , 400 , 800 , respectively, the R software is adopted to compute f n ( x ) with f ( x ) = cos 2 π x and f ( x ) = x 2 − 2 sin x for 1,000 times. The asymptotic results are given in Figures 1, 2, 3 and Figures 4, 5, 6, respectively. From Figures 1–3 and Figures 4–6, we can see that under the aforementioned two cases, the estimators converge to the true functions as n increases.

$Figure 1 Comparison of f n ( x ) {f}_{n}\left(x) and f ( x ) = cos ( 2 π x ) f\left(x)=\cos \left(2\pi x) with n = 200 n=200 .$

Figure 1

Comparison of f n ( x ) and f ( x ) = cos ( 2 π x ) with n = 200 .

$Figure 2 Comparison of f n ( x ) {f}_{n}\left(x) and f ( x ) = cos ( 2 π x ) f\left(x)=\cos \left(2\pi x) with n = 400 n=400 .$

Figure 2

Comparison of f n ( x ) and f ( x ) = cos ( 2 π x ) with n = 400 .

$Figure 3 Comparison of f n ( x ) {f}_{n}\left(x) and f ( x ) = cos ( 2 π x ) f\left(x)=\cos \left(2\pi x) with n = 600 n=600 .$

Figure 3

Comparison of f n ( x ) and f ( x ) = cos ( 2 π x ) with n = 600 .

$Figure 4 Comparison of f n ( x ) {f}_{n}\left(x) and f ( x ) = x 2 2 sin x f\left(x)={x}^{2}2\sin x with n = 200 n=200 .$

Figure 4

Comparison of f n ( x ) and f ( x ) = x 2 2 sin x with n = 200 .

$Figure 5 Comparison of f n ( x ) {f}_{n}\left(x) and f ( x ) = x 2 2 sin x f\left(x)={x}^{2}2\sin x with n = 400 n=400 .$

Figure 5

Comparison of f n ( x ) and f ( x ) = x 2 2 sin x with n = 400 .

$Figure 6 Comparison of f n ( x ) {f}_{n}\left(x) and f ( x ) = x 2 2 sin x f\left(x)={x}^{2}2\sin x with n = 800 n=800 .$

Figure 6

Comparison of f n ( x ) and f ( x ) = x 2 2 sin x with n = 800 .

Now, we turn to verify the numerical performance of the estimators β ˆ n 1 and β ˆ n 2 in Example 2.1. The sample { ε n , n ≥ 1 } is also generated by (3.1) with θ n = n − 1 . Then, it is easy to check by Wang et al. [1] that g ( n ) = O ( n ) . Choosing n = 100 , 200 , 300 , and 400, respectively, the R software is used to compute β ˆ n 1 and β ˆ n 2 for 1,000 times. The average bias and the root mean-squared errors (RMSE) of the two estimators are provided in Table 1. Table 1 reveals that the average bias converges to zero, and the fluctuation range decreases markedly as n increases. These aforementioned simulation results agree with the main results established in Section 2.

Table 1

Average bias and RMSE of β ˆ n 1 and β ˆ n 2

Estimators	n	Average bias	RMSE
	100	1.55 × 1 0 ‒ 3	0.1049
β ˆ n 1	200	‒ 9.25 × 1 0 ‒ 4	0.0728
	300	8.13 × 1 0 ‒ 4	0.0583
	400	‒ 8.42 × 1 0 ‒ 4	0.051
	100	3.62 × 1 0 ‒ 5	0.004
β ˆ n 2	200	‒ 3.07 × 1 0 ‒ 5	0.0014
	300	‒ 1.07 × 1 0 ‒ 5	7.69 × 1 0 ‒ 4
	400	8.38 × 1 0 ‒ 6	5.07 × 1 0 ‒ 4

4 Proofs of the results

Before proving our main results, we first give two basic lemmas, the first one of which is presented by Wang et al. [9].

Lemma 4.1

Let { X n , n ≥ 1 } be a sequence of WOD random variables. If { f n ( ⋅ ) , n ≥ 1 } are all nondecreasing (or nonincreasing) real functions, then { f n ( X n ) , n ≥ 1 } are still WOD.

The next lemma concerning the Bernstein-type inequality for WOD random variables was proved in the study of Xia et al. [13].

Lemma 4.2

Let { X n , n ≥ 1 } be a sequence of WOD random variables with E X n = 0 and ∣ X n ∣ ≤ d n a.s . for each n ≥ 1 , where { d n , n ≥ 1 } is a sequence of positive constants. Denote b n = max 1 ≤ i ≤ n d i and Δ n 2 = ∑ i = 1 n E X i 2 for each n ≥ 1 . Then, for every ε > 0 ,

P ∑ i = 1 n X i ≥ ε ≤ 2 g ( n ) exp − ε 2 2 ( 2 Δ n 2 + b n ε ) .

With the aforementioned lemmas, we now present the proofs of the main results as follows.

Proof of Theorem 2.1

Set 0 ≤ ϱ < q < ( p − α ) ⁄ ( α p − α ) . Denote for fixed n ≥ 1 , 1 ≤ i ≤ n that

X n , i ( 1 ) = − n − q I ( a n i X i < − n − q ) + a n i X i I ( ∣ a n i X i ∣ ≤ n − q ) + n − q I ( a n i X i > n − q ) ; X n , i ( 2 ) = ( a n i X i − n − q ) I ( a n i X i > n − q ) + ( a n i X i + n − q ) I ( a n i X i < − n − q ) .

By Lemma 4.1, we can see that { X n , i ( 1 ) , 1 ≤ i ≤ n , n ≥ 1 } is still an array of WOD random variables. It follows from E X n = 0 and q < ( p − α ) ⁄ ( α p − α ) ≤ 1 ⁄ α that under condition ( i ) ,

∑ i = 1 n E X n , i ( 1 ) ≤ ∑ i = 1 n E ∣ a n i X i ∣ I ( ∣ a n i X i ∣ > n − q ) ≤ n ( p − 1 ) q ∑ i = 1 n a n i 2 ⋅ ∣ a n i ∣ p − 2 sup i ≥ 1 E ∣ X i ∣ p ≤ C n ( p − 1 ) q − ( p − 2 ) ⁄ α log − 1 n ≤ C n ( 2 − α ) ⁄ α log − 1 n → 0 as n → ∞ ;

and under condition ( i i ) ,

∑ i = 1 n E X n , i ( 1 ) ≤ ∑ i = 1 n E ∣ a n i X i ∣ I ( ∣ a n i X i ∣ > n − q ) ≤ n ( p − 1 ) q ∑ i = 1 n ∣ a n i ∣ ⋅ ∣ a n i ∣ p − 1 sup i ≥ 1 E ∣ X i ∣ p ≤ C n ( p − 1 ) ( q − 1 ⁄ α ) → 0 , as n → ∞ .

Note that ∣ X n , i ( 1 ) − E X n , i ( 1 ) ∣ ≤ 2 n − q for each 1 ≤ i ≤ n . On the other hand, by condition ( i ) , we have that ∑ i = 1 n E ∣ X n , i ( 1 ) − E X n , i ( 1 ) ∣ 2 ≤ ∑ i = 1 n a n i 2 E X i 2 = o ( log − 1 n ) ; from condition ( i i ) , we can also see if p ≥ 2 , ∑ i = 1 n E ∣ X n , i ( 1 ) − E X n , i ( 1 ) ∣ 2 ≤ ∑ i = 1 n ∣ a n i ∣ ⋅ max 1 ≤ i ≤ n ∣ a n i ∣ = o ( n − q ) , and if 1 < p < 2 , ∑ i = 1 n E ∣ X n , i ( 1 ) − E X n , i ( 1 ) ∣ 2 ≤ n − q ( 2 − p ) ∑ i = 1 n ∣ a n i ∣ ⋅ max 1 ≤ i ≤ n ∣ a n i ∣ p − 1 E ∣ X i ∣ p = o ( n − q ) . Hence, we obtain by Lemma 4.2 that under condition ( i ) ,

∑ n = 1 ∞ P ∑ i = 1 n X n , i ( 1 ) > ε ≤ C ∑ n = 1 ∞ P ∑ i = 1 n ( X n , i ( 1 ) − E X n , i ( 1 ) ) > ε ⁄ 2 ≤ C ∑ n = 1 ∞ g ( n ) exp − ε 2 ⁄ 4 2 [ o ( log − 1 n ) + ε n − q ] ≤ C ∑ n = 1 ∞ n τ exp { − ( τ + 2 ) log n } ≤ C ∑ n = 1 ∞ n − 2 < ∞ ;

and under condition ( i i ) ,

∑ n = 1 ∞ P ∑ i = 1 n X n , i ( 1 ) > ε ≤ C ∑ n = 1 ∞ P ∑ i = 1 n ( X n , i ( 1 ) − E X n , i ( 1 ) ) > ε ⁄ 2 ≤ C ∑ n = 1 ∞ g ( n ) exp − ε 2 ⁄ 4 2 [ o ( n − q ) + ε n − q ] ≤ C ∑ n = 1 ∞ exp { n ϱ − ε n q ⁄ 16 } < ∞ ,

which together with Borel-Cantelli lemma yields that ∑ i = 1 n X n , i ( 1 ) → 0 a.s. Now, we turn to prove ∑ i = 1 n X n , i ( 2 ) → 0 a . s . Note by q < ( p − α ) ⁄ ( α p − α ) that

∑ i = 1 ∞ i − 1 ⁄ α E ∣ X i ∣ I ( ∣ X i ∣ > C i 1 ⁄ α − q ) ≤ C ∑ i = 1 ∞ i − 1 ⁄ α ⋅ i ( 1 ⁄ α − q ) ( 1 − p ) sup i ≥ 1 E ∣ X i ∣ p < ∞ ,

which implies

∑ i = 1 ∞ i − 1 ⁄ α ∣ X i ∣ I ( ∣ X i ∣ > C i 1 ⁄ α − q ) < ∞ a.s .

Therefore, we can derive by the Kronecker’s lemma that

∑ i = 1 n X n , i ( 2 ) ≤ ∑ i = 1 n ∣ a n i X i ∣ I ( ∣ a n i X i ∣ > n − q ) ≤ C n − 1 ⁄ α ∑ i = 1 n ∣ X i ∣ I ( ∣ X i ∣ > C n 1 ⁄ α − q ) ≤ C n − 1 ⁄ α ∑ i = 1 n ∣ X i ∣ I ( ∣ X i ∣ > C i 1 ⁄ α − q ) → 0 a.s .

The proof is complete.□

Proof of Theorem 2.2

The method is similar to the proof of Theorem 2.1. Set 1 ⁄ λ < q < 1 ⁄ α − ( 1 ⁄ λ − 1 ⁄ α + 1 ) ⁄ ( p − 1 ) . Denote for fixed n ≥ 1 , 1 ≤ i ≤ n that

Y n , i ( 1 ) = − n − q I ( a n i X i < − n − q ) + a n i X i I ( ∣ a n i X i ∣ ≤ n − q ) + n − q I ( a n i X i > n − q ) ; Y n , i ( 2 ) = ( a n i X i − n − q ) I ( a n i X i > n − q ) + ( a n i X i + n − q ) I ( a n i X i < − n − q ) .

It follows from Lemma 4.1 that { Y n , i ( 1 ) , 1 ≤ i ≤ n , n ≥ 1 } is still an array of WOD random variables. Hence, we have by E X n = 0 and q < 1 ⁄ α − ( 1 ⁄ λ − 1 ⁄ α + 1 ) ⁄ ( p − 1 ) that

n 1 ⁄ λ ∑ i = 1 n E Y n , i ( 1 ) ≤ n 1 ⁄ λ ∑ i = 1 n E ∣ a n i X i ∣ I ( ∣ a n i X i ∣ > n − q ) ≤ n 1 ⁄ λ + ( p − 1 ) q ∑ i = 1 n a n i 2 ⋅ ∣ a n i ∣ p − 2 sup i ≥ 1 E ∣ X i ∣ p ≤ C n − 1 ⁄ λ + ( p − 1 ) q − ( p − 2 ) ⁄ α log − 1 n ≤ C n 2 ⁄ α − 2 ⁄ λ − 1 log − 1 n → 0 as n → ∞ ,

i.e., to say, ∑ i = 1 n E Y n , i ( 1 ) ≤ D n − 1 ⁄ λ ⁄ 2 for any D > 0 . On the other hand, note that ∣ Y n , i ( 1 ) − E Y n , i ( 1 ) ∣ ≤ 2 n − q and ∑ i = 1 n E ∣ Y n , i ( 1 ) − E Y n , i ( 1 ) ∣ 2 ≤ ∑ i = 1 n a n i 2 E X i 2 ≤ C n − 2 ⁄ λ log − 1 n for each 1 ≤ i ≤ n . Hence, we obtain by Lemma 4.2 again and choosing D > 24 C ( τ + 1 ) that

∑ n = 1 ∞ P ∑ i = 1 n Y n , i ( 1 ) > D n − 1 ⁄ λ ≤ C ∑ n = 1 ∞ P ∑ i = 1 n ( Y n , i ( 1 ) − E Y n , i ( 1 ) ) > D n − 1 ⁄ λ ⁄ 2 ≤ C ∑ n = 1 ∞ g ( n ) exp − D 2 n − 2 ⁄ λ ⁄ 4 2 [ 2 C n − 2 ⁄ λ log − 1 n + D n − 1 ⁄ λ − q ] ≤ C ∑ n = 1 ∞ g ( n ) exp − D 2 n − 2 ⁄ λ ⁄ 4 2 [ 3 C n − 2 ⁄ λ log − 1 n ] ≤ C ∑ n = 1 ∞ n τ − D 2 ⁄ ( 24 C ) < ∞ ,

which together with Borel-Cantelli lemma yields that ∣ ∑ i = 1 n Y n , i ( 1 ) ∣ = O ( n − 1 ⁄ λ ) a.s. Now, we turn to prove ∣ ∑ i = 1 n Y n , i ( 2 ) ∣ = O ( n − 1 ⁄ λ ) a.s . Note by q < 1 ⁄ α − ( 1 ⁄ λ − 1 ⁄ α + 1 ) ⁄ ( p − 1 ) that

∑ i = 1 ∞ i 1 ⁄ λ − 1 ⁄ α E ∣ X i ∣ I ( ∣ X i ∣ > C i 1 ⁄ α − q ) ≤ C ∑ i = 1 ∞ i 1 ⁄ λ − 1 ⁄ α ⋅ i ( 1 ⁄ α − q ) ( 1 − p ) sup i ≥ 1 E ∣ X i ∣ p < ∞ ,

which implies

∑ i = 1 ∞ i 1 ⁄ λ − 1 ⁄ α ∣ X i ∣ I ( ∣ X i ∣ > C i 1 ⁄ α − q ) < ∞ a.s .

Hence, we can derive by the Kronecker’s lemma that

n 1 ⁄ λ ∑ i = 1 n Y n , i ( 2 ) ≤ n 1 ⁄ λ ∑ i = 1 n ∣ a n i X i ∣ I ( ∣ a n i X i ∣ > n − q ) ≤ C n 1 ⁄ λ − 1 ⁄ α ∑ i = 1 n ∣ X i ∣ I ( ∣ X i ∣ > C n 1 ⁄ α − q ) ≤ C n 1 ⁄ λ − 1 ⁄ α ∑ i = 1 n ∣ X i ∣ I ( ∣ X i ∣ > C i 1 ⁄ α − q ) → 0 a.s .

The proof is complete.□

Proof of Theorem 2.3

For a > 0 and x ∈ c ( f ) , it follows from (2.1) and (2.2) that

(4.1) ∣ E f n ( x ) − f ( x ) ∣ ≤ ∑ i = 1 n ∣ ω n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ ≤ a ) + ∑ i = 1 n ∣ ω n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ > a ) + ∣ f ( x ) ∣ ⋅ ∑ i = 1 n ω n i ( x ) − 1 .

By x ∈ c ( f ) , we derive that for all ε > 0 , there exists a positive constant δ such that for all x 0 satisfying ‖ x 0 − x ‖ < δ , ∣ f ( x 0 ) − f ( x ) ∣ < ε . Setting 0 < a < δ in (4.1), we obtain that

∣ E f n ( x ) − f ( x ) ∣ ≤ ε ∑ i = 1 n ∣ ω n i ( x ) ∣ + ∑ i = 1 n ∣ ω n i ( x ) ∣ ⋅ ∣ f ( x n i ) − f ( x ) ∣ I ( ‖ x n i − x ‖ > a ) + ∣ f ( x ) ∣ ⋅ ∑ i = 1 n ω n i ( x ) − 1 .

Therefore, it follows from assumptions ( H 1 ) – ( H 3 ) and the arbitrariness of ε > 0 that for all x ∈ c ( f ) ,

(4.2) lim n → ∞ E f n ( x ) = f ( x ) .

In view of (4.2), to obtain the desired result (2.3), it suffices to prove

(4.3) f n ( x ) − E f n ( x ) = ∑ i = 1 n ω n i ( x ) ε n i → 0 a.s.

Since the joint distribution of ( ε n 1 , ε n 2 , … , ε n n ) is the same as ( ε 1 , ε 2 , … , ε n ) , we have that for any ε > 0 ,

(4.4) P ∑ i = 1 n ω n i ( x ) ε n i > ε , i.o. = P ∑ i = 1 n ω n i ( x ) ε i > ε , i.o. .

Applying Theorem 2.1 with X i = ε i and a n i = ω n i ( x ) , we can obtain

∑ i = 1 n ω n i ( x ) ε i → 0 a.s. ,

which together with (4.4) gets the desired result (4.3) immediately. This completes the proof of the theorem.□

Proof of Theorem 2.4

In view of (2.6), we only need to prove

∑ i = 1 n b n i ( j ) b n ( j ) ε i → 0 , a.s. , 1 ≤ j ≤ p .

Set a n i = b n i ( j ) b n ( j ) . It can be easily checked by the conditions of Theorem 2.5 that max 1 ≤ i ≤ n ∣ a n i ∣ = O ( n − 1 ⁄ α ) , and

∑ i = 1 n a n i 2 = ∑ i = 1 n ( b n i ( j ) ) 2 ⁄ ( b n ( j ) ) 2 = 1 ⁄ b n ( j ) = O ( n − 2 ⁄ λ log − 1 n ) .

Therefore, the desired result follows from Theorem 2.2 immediately. This completes the proof of the theorem.□

Acknowledgements

The authors are most grateful to the editor and anonymous referees for carefully reading the manuscript and for valuable suggestions that helped in improving an earlier version of this manuscript.

Funding information: This study was supported by the Outstanding Youth Research Project of Anhui Colleges (2022AH030156).
Author contributions: YZ: writing original draft; WW: data curation; KC: review and supervision. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare no conflict of interest.
Data availability statement: Not applicable.

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Received: 2023-05-17

Revised: 2024-03-31

Accepted: 2024-06-06

Published Online: 2024-08-26

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https://doi.org/10.1515/math-2024-0027

Keywords for this article

strong law of large numbers; weighted sums; widely orthant dependent; strong consistency

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