Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model

Wenjing Hu; Wei Wang; Yi Wu

doi:10.1515/math-2024-0003

Article Open Access

Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model

Wenjing Hu , Wei Wang and Yi Wu

Published/Copyright: May 22, 2024

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$Open Mathematics$

From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, the complete convergence and the Kolmogorov strong law of large numbers for weighted sums of ( α , β ) -mixing random variables are presented. An application to simple linear errors-in-variables model is provided. Simulation studies are also carried out to support the theoretical results.

Keywords: (α, β)-mixing random variables; complete convergence; strong law of large numbers; simple linear errors-in-variables model

MSC 2010: 60F15; 62G05

1 Introduction

In the past decades, the dependence assumption in probability theory and statistical applications is becoming more and more common since the independence assumption exhibits its implausible in many related fields. Actually, many dependence structures have been raised and carefully investigated. Among various dependence structures, ( α , β ) -mixing is quite general but less studied. In this article, we will study the complete convergence and the Kolmogorov strong law of large numbers (SLLN) for weighted sums of ( α , β ) -mixing random variables. Now, we recall the concept of ( α , β ) -mixing random variables as follows:

Definition 1.1

Let { X n , n ≥ 1 } be a sequence of random variables or random vectors defined on a fixed probability space ( Ω , ℱ , P ). Let n and m be positive integers. Write ℱ n m = σ ( X i , n ≤ i ≤ m ) . Given σ -algebras A , ℬ in ℱ , let

λ ( A , ℬ ) = sup X ∈ L 1 ∕ α ( A ) , Y ∈ L 1 ∕ β ( ℬ ) ∣ E X Y − E X E Y ∣ ‖ X ‖ 1 ∕ α ‖ Y ‖ 1 ∕ β ,

where ‖ X ‖ p = ( E ∣ X ∣ p ) 1 ∕ p . Define the ( α , β )-mixing coefficients by

λ ( n ) = sup k ≥ 1 λ ( ℱ 1 k , ℱ k + n ∞ ) , n ≥ 0 .

Let 0 < α , β < 1 , and α + β = 1 . The sequence { X n , n ≥ 1 } is said to be ( α , β )-mixing if λ ( n ) ↓ 0 as n → ∞ .

Since the concept of ( α , β ) -mixing random variables was proposed by Bradley and Bryc [1], many authors studied its limit theorems and applications. For example, Cai [2] obtained strong consistency and rates for recursive nonparametric conditional probability density estimates under ( α , β ) -mixing conditions; Shen et al. [3] studied some convergence theorems for ( α , β ) -mixing random variables and obtained some new SLLN; Yang [4] studied the convergence properties through the analysis of ( α , β ) -mixing random variables in different situations; Qi [5] studied SLLN of partial sums and sums of products for ( α , β ) -mixing sequence; Gao [6] investigated the ( α , β )-mixing sequences, which are stochastically dominated, and presented some strong stability; Yu [7] showed the Rosenthal-type inequality of the ( α , β )-mixing sequences and investigated the strong convergence theorems; Samura et al. [8] investigated the strong consistency, complete consistency, and mean consistency for the estimators of partially linear regression models under ( α , β )-mixing errors. It is well known that the complete convergence not only reveals the SLLN but also characterizes the rate of convergence. Therefore, since the concept of complete convergence was proposed by Hsu and Robbins [9], many corresponding results have been established successively. For example, Ghosal and Chandra [10] studied complete convergence of martingale arrays under rather weak conditions; Chen [11] established SLLN and complete convergence for weighted sums of negatively associated (NA) random variables under certain moment conditions; Liang and Su [12] obtained the complete convergence for weighted sums of NA sequence under general weighted coefficients; Sung [13] obtained the complete convergence for weighted sums of ρ * -mixing random variables; Shen et al. [14] studied the complete convergence and the complete moment convergence for extended negatively dependent random variables without identical distribution; Wu et al. [15] studied the strong convergence for weighted sums of widely orthant dependent random variables.

Inspired by the aforementioned literature, especially Wu et al. [15], we will study the complete convergence and the Kolmogorov SLLN for weighted sums of ( α , β ) -mixing random variables. As an application, the sufficient and necessary condition of strong consistency for least squares estimators in a simple linear errors-in-variables (EV) model is provided. There are already some investigations on the simple linear EV models. For example, Miao et al. [16] obtained the central limit theorems for least squares estimators in simple linear EV regression model under some mild conditions; Fan et al. [17] studied the asymptotic behavior of least squares estimators in the EV regression model with correlation error; Zhang et al. [18] studied the weak consistency and convergence rate of least squares estimators in EV models with independent and identically distributed errors; Peng et al. [19] proved the necessary and sufficient condition of least squares estimator in EV regression models with the extended negatively dependent errors; Wu et al. [20] proved the necessary and sufficient condition of least squares estimators in EV regression models with the widely orthant dependent errors, which extends the corresponding results of Chen et al. [21] for independent errors.

In this article, the complete convergence and the Kolmogorov SLLN for weighted sums of ( α , β ) -mixing random variables are investigated, which are not obtained before. As an application, the necessary and sufficient condition of least squares estimators in a simple linear EV regression models is proved. These results are extensions of some existing ones. Simulation studies are also carried out to support the theoretical results.

This article is laid out as follows. The main conclusions are provided in Section 2. Section 3 provides some important lemmas, and Section 4 gives proofs of the main conclusions. In this document, C represents some positive constant whose value may vary in different places. Let log x = ln max ( x , e ) and I ( A ) be the indicator function of the event A . Denote x + = x I ( x ≥ 0 ) and x − = − x I ( x < 0 ) . a n = o ( b n ) implies a n ∕ b n → 0 and a n = O ( b n ) means sup n ≥ 1 a n ∕ b n < ∞ .

2 Main results

2.1 Strong convergence

Now we state our main results, the first one of which is the complete convergence for weighted sums of ( α , β )-mixing random variables.

Theorem 2.1

Let r > 1 , 1 ≤ p < 2 , s > p , t > p with 1 ∕ s + 1 ∕ t = 1 ∕ p . Let { X , X n , n ≥ 1 } be a sequence of identically distributed ( α , β )-mixing random variables with ∑ n = 1 ∞ ( λ ( n ) ) 1 2 α ∧ 1 2 β < ∞ , where 0 < α , β < 1 with α + β = 1 , { a n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of constants satisfying

(2.1) ∑ i = 1 n ∣ a n i ∣ s = O ( n ) .

E X = 0 , E ∣ X ∣ ( r − 1 ) t < ∞ , i f s < r p , E ∣ X ∣ ( r − 1 ) t log ∣ X ∣ < ∞ , i f s = r p , E ∣ X ∣ r p < ∞ , i f s > r p ,

then for any ε > 0 ,

(2.2) ∑ n = 1 ∞ n r − 2 P ∑ i = 1 n a n i X i > ε n 1 ∕ p < ∞ .

For r = p = 1 , we can obtain the following result for maximum weighted sums of ( α , β )-mixing random variables.

Theorem 2.2

Let { X , X n , n ≥ 1 } be a sequence of ( α , β )-mixing random variables with E X = 0 and ∑ n = 1 ∞ ( λ ( n ) ) 1 2 α ∧ 1 2 β < ∞ , where 0 < α , β < 1 with α + β = 1 , and { a n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of constants satisfying (2.1) for some s > 1 . Then for any ε > 0 ,

(2.3) ∑ n = 1 ∞ n − 1 P max 1 ≤ j ≤ n ∑ i = 1 j a n i X i > ε n < ∞ .

Taking a n i = a i for each 1 ≤ i ≤ n , n ≥ 1 in Theorem 2.2, we can obtain the following single-indexed weighted version of Kolmogorov SLLN for ( α , β )-mixing random variables by some standard calculation.

Corollary 2.1

Let { X , X n , n ≥ 1 } be a sequence of ( α , β )-mixing random variables with E X = 0 and ∑ n = 1 ∞ ( λ ( n ) ) 1 2 α ∧ 1 2 β < ∞ , where 0 < α , β < 1 with α + β = 1 , and { a n , n ≥ 1 } be a sequence of constants satisfying ∑ i = 1 n ∣ a i ∣ s = O ( n ) for some s > 1 . Then

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

By virtue of Corollary 2.1, we can further obtain the following double-indexed weighted version of Kolmogorov SLLN for ( α , β )-mixing random variables.

Corollary 2.2

Let s > 1 , t > 1 with 1 ∕ s + 1 ∕ t = 1 . Suppose that { X , X n , n ≥ 1 } is a sequence of ( α , β )-mixing random variables with ∑ n = 1 ∞ ( λ ( n ) ) 1 2 α ∧ 1 2 β < ∞ , where 0 < α , β < 1 with α + β = 1 . Let { a n i , 1 ≤ i ≤ n , n ≥ 1 } be a sequence of constants satisfying ∑ i = 1 n ∣ a n i ∣ s = O ( n ) for some s > 1 , where for s = ∞ , we interpret it as sup 1 ≤ i ≤ n ∣ a n i ∣ < ∞ . If E X = 0 and E ∣ X ∣ t < ∞ , then

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

2.2 Application to linear EV model

Consider the following linear EV regression model which was raised by Deaton [22]:

(2.4) η i = θ 0 + β 0 x i + ε i , ξ i = x i + δ i , 1 ≤ i ≤ n ,

where θ 0 , β 0 , x 1 , x 2 , … are unknown constants (parameters), ( ε 1 , δ 1 ) , ( ε 2 , δ 2 ) , … are random vectors and ξ i , η i , i = 1 , 2 , … are observable. From (2.4), we have

(2.5) η i = θ 0 + β 0 ξ i + ν i , ν i = ε i − β 0 δ i , 1 ≤ i ≤ n .

Considering formula (2.5) as a usual regression model of η i on ξ i , we obtain the least squares (LS) estimators of β 0 and θ 0 as follows:

β ˆ n = ∑ i = 1 n ( ξ i − ξ ¯ n ) ( η i − η ¯ n ) ∑ i = 1 n ( ξ i − ξ ¯ n ) 2 , θ ˆ n = η ¯ n − β ˆ n ξ ¯ n ,

where ξ ¯ n = n − 1 ∑ i = 1 n ξ i , and other similar notations, such as η ¯ n , δ ¯ n , and x ¯ n , are all defined in the same way. Denote S n = ∑ i = 1 n ( x i − x ¯ n ) 2 for each n ≥ 1 . On the basis of the aforementioned notations, we can obtain that

(2.6) β ˆ n − β 0 = ∑ i = 1 n ( δ i − δ ¯ n ) ε i + ∑ i = 1 n ( x i − x ¯ n ) ( ε i − β 0 δ i ) − β 0 ∑ i = 1 n ( δ i − δ ¯ n ) 2 ∑ i = 1 n ( ξ i − ξ ¯ n ) 2

and

(2.7) θ ˆ n − θ 0 = ( β 0 − β ˆ n ) x ¯ n + ( β 0 − β ˆ n ) δ ¯ n + ε ¯ n − β 0 δ ¯ n .

EV model is simple in form and widely used, so the study of EV model has received extensive attention. More relevant research can be referred to Amemiya and Fuller [23], Cui and Chen [24], Hu et al. [25], Shen [26], Yi et al. [27], Wang and Wang [28] and the references therein.

Before presenting our result, the following assumptions are indispensable.

( A 1 ) { ( ε i , δ i ) , i ≥ 1 } is a sequence of identically distributed ( α , β ) -mixing random vectors with ∑ n = 1 ∞ ( λ ( n ) ) 1 2 α ∧ 1 2 β < ∞ , where 0 < α , β < 1 with α + β = 1 .

( A 2 ) E ε 1 = 0 , E δ 1 = 0 , E ε 1 2 < ∞ , and E δ 1 2 < ∞ .

Theorem 2.3

Under the model (2.4), suppose that ( A 1 ) and ( A 2 ) hold. Then S n ∕ n → ∞ implies

β ˆ n → β 0 a.s.

Conversely, if E ε 1 δ 1 − β 0 E δ 1 2 ≠ 0 , the β ˆ n → β 0 a.s. implies S n ∕ n → ∞ as n → ∞ .

Theorem 2.4

Under the model (2.4), suppose that ( A 1 ) and ( A 2 ) hold. Assume further that sup n ≥ 1 n x ¯ n 2 ∕ S n * < ∞ , where S n * = max ( n , S n ) . Then n x ¯ n ∕ S n * → 0 implies

θ ˆ n → θ 0 a . s .

Conversely, if E ε 1 δ 1 − β 0 E δ 1 2 ≠ 0 , then θ ˆ n → θ 0 a . s . implies n x ¯ n ∕ S n * as n → ∞ .

2.3 Numerical simulation

In this subsection, we will carry out a simple simulation to study the numerical performance of the LS estimators β ˆ n and θ ˆ n . First, we simulate the consistent result established in Theorems 2.3 and 2.4. The data are generated from (2.4).

For fixed positive integer m , let e i ∼ i . i . d N ( 0 , σ 0 2 ) , where σ 0 2 = 1 ∕ ( m + 1 ) and m = 10 . Let ε i = ∑ k = 0 m e i + k for 1 ≤ i ≤ n . Then { ε i , i ≥ 1 } is a sequence of ( α , β ) -mixing random variables with ε i ∼ N ( 0 , 1 ) for each i ≥ 1 . Set x i = i ∕ n 0.3 for all 1 ≤ i ≤ n , sample size n as n = 100 , 200 , 300 , 500 . δ i , 1 ≤ i ≤ n are generated in the same way. For different values of β and θ , we use the Python to calculate β ˆ n and θ ˆ n for 1,000 times to obtain the Violin-plots in Figures 1, 2, 3, 4, 5, 6. We can see that as the sample size n increases, the fitting effect becomes better. This confirms the validity of Theorems 2.3 and 2.4.

$Figure 1 Violin-plots of β ˆ n ‒ β {\hat{\beta }}_{n}‒\beta with β = 2 \beta =2 and θ = 3 \theta =3 .$

Figure 1

Violin-plots of β ˆ n ‒ β with β = 2 and θ = 3 .

$Figure 2 Violin-plots of θ ˆ n ‒ θ {\hat{\theta }}_{n}‒\theta with β = 2 \beta =2 and θ = 3 \theta =3 .$

Figure 2

Violin-plots of θ ˆ n ‒ θ with β = 2 and θ = 3 .

$Figure 3 Violin-plots of β ˆ n ‒ β {\hat{\beta }}_{n}‒\beta with β = 3 \beta =3 and θ = 2 \theta =2 .$

Figure 3

Violin-plots of β ˆ n ‒ β with β = 3 and θ = 2 .

$Figure 4 Violin-plots of θ ˆ n ‒ θ {\hat{\theta }}_{n}‒\theta with β = 3 \beta =3 and θ = 2 \theta =2 .$

Figure 4

Violin-plots of θ ˆ n ‒ θ with β = 3 and θ = 2 .

$Figure 5 Violin-plots of β ˆ n ‒ β {\hat{\beta }}_{n}‒\beta with β = 4 \beta =4 and θ = 5 \theta =5 .$

Figure 5

Violin-plots of β ˆ n ‒ β with β = 4 and θ = 5 .

$Figure 6 Violin-plots of θ ˆ n ‒ θ {\hat{\theta }}_{n}‒\theta with β = 4 \beta =4 and θ = 5 \theta =5 .$

Figure 6

Violin-plots of θ ˆ n ‒ θ with β = 4 and θ = 5 .

Case 1: β = 2 and θ = 3 . By taking the sample size n as n = 100 , 200 , 300 , 500 respectively, we compute β ˆ n − β and θ ˆ n − θ for 1,000 times to obtain the Violin-plots in Figures 1 and 2.

Case 2: β = 3 and θ = 2 . Computed β ˆ n − β and θ ˆ n − θ for 1,000 times to obtain the Violin-plots in Figures 3 and 4.

Case 3: β = 4 and θ = 5 . We also compute β ˆ n − β and θ ˆ n − θ for 1,000 times to obtain the Violin-plots in Figures 5 and 6.

In Figures 1–6, we can see clearly that with different values of β and θ , both β ˆ n − β and θ ˆ n − θ tend to the zero line and the fluctuation ranges of β ˆ n − β and θ ˆ n − θ decrease as n increases, and the fitting effect improves. These validate the consistency results we have established.

In addition, we calculated the mean square error (MSE) and absolute bias (ABias) of β and θ , and the results are obtained in Table 1. We can conclude that when the values of β and θ are determined, there is a bias when the sample size is small, but as the sample size increases, the bias as well as MSE gradually decrease. These show a good fit of our theoretical results.

Table 1

MSE and ABias of β ˆ n and θ ˆ n

Estimator	( β , θ )	MSE/ABias	n = 100	n = 200	n = 300	n = 500
β ˆ n	(2,3)	MSE	9.21 × 1 0 ‒ 3	3.71 × 1 0 ‒ 4	1.33 × 1 0 ‒ 4	4.35 × 1 0 ‒ 6
		ABias	7.64 × 1 0 ‒ 2	1.52 × 1 0 ‒ 2	9.45 × 1 0 ‒ 3	5.27 × 1 0 ‒ 3
	(3,2)	MSE	1.79 × 1 0 ‒ 2	3.66 × 1 0 ‒ 4	1.43 × 1 0 ‒ 4	4.04 × 1 0 ‒ 5
		ABias	1.06 × 1 0 ‒ 1	1.53 × 1 0 ‒ 2	9.68 × 1 0 ‒ 3	5.09 × 1 0 ‒ 3
	(4,5)	MSE	2.88 × 1 0 ‒ 2	3.75 × 1 0 ‒ 4	1.36 × 1 0 ‒ 4	4.56 × 1 0 ‒ 5
		ABias	1.37 × 1 0 ‒ 1	1.53 × 1 0 ‒ 2	9.31 × 1 0 ‒ 3	5.36 × 1 0 ‒ 3
θ ˆ n	(2,3)	MSE	2.04	0.16	0.10	0.07
		ABias	1.14	0.31	0.26	0.20
	(3,2)	MSE	3.91	0.15	0.11	0.06
		ABias	1.57	0.31	0.26	0.19
	(4,5)	MSE	6.19	0.16	0.10	0.07
		ABias	2.02	0.31	0.25	0.21

3 Some important lemmas

Several lemmas to prove the main results of the article are stated in this section. The first one is the Rosenthal type inequality for weighted sums of ( α , β )-mixing random variables, which can be found in the study by Yu [7].

Lemma 3.1

Let { X n , n ≥ 1 } be a sequence of ( α , β )-mixing random variables with E X n = 0 , E ∣ X n ∣ p < ∞ , and ∑ n = 1 ∞ ( λ ( n ) ) 1 2 α ∧ 1 2 β < ∞ , where p ≥ 2 , 0 < α , β < 1 , and α + β = 1 . Assume that { a n i , 1 ≤ i ≤ n , n ≥ 1 } is an array of real numbers. Then there exists a positive constant C depending only on α , β , and λ ( ⋅ ) such that

E ∑ i = 1 n a n i X i p ≤ C ∑ i = 1 n ∣ a n i ∣ p E ∣ X i ∣ p + ∑ i = 1 n a n i 2 E X i 2 p ∕ 2 .

By adopting the method used in Theorem 2.3.1 of Stout [29], we can obtain the following Rosenthal type maximum inequality by Lemma 3.1. The details are omitted here.

Lemma 3.2

Let { X n , n ≥ 1 } be a sequence of ( α , β )-mixing random variables with E X n = 0 , E ∣ X n ∣ p < ∞ and ∑ n = 1 ∞ ( λ ( n ) ) 1 2 α ∧ 1 2 β < ∞ , where p ≥ 2 , 0 < α , β < 1 , and α + β = 1 . Assume that { a n i , 1 ≤ i ≤ n , n ≥ 1 } is an array of real numbers. Then there exists a positive constant C depending only on α , β , and λ ( ⋅ ) such that

E max 1 ≤ j ≤ n ∑ i = 1 j a n i X i p ≤ C ( log n ) p ∑ i = 1 n ∣ a n i ∣ p E ∣ X i ∣ p + ∑ i = 1 n a n i 2 E X i 2 p ∕ 2 .

The following two lemmas can be found in the study by Chen and Sung [30].

Lemma 3.3

Let r ≥ 1 , 0 < p < 2 , s > 0 , t > 0 with 1 ∕ s + 1 ∕ t = 1 ∕ p , and let X be a random variable. Let { a n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of constants satisfying sup n ≥ 1 n − 1 ∑ i = 1 n ∣ a n i ∣ s < ∞ . Then

∑ n = 1 ∞ n r − 2 ∑ i = 1 n P { ∣ a n i X ∣ > n 1 ∕ p } ≤ C E ∣ X ∣ ( r − 1 ) t , i f s < r p , C E ∣ X ∣ ( r − 1 ) t log ( 1 + ∣ X ∣ ) , i f s = r p , C E ∣ X ∣ r p , i f s > r p .

Lemma 3.4

Let r ≥ 1 , 0 < p < 2 , s > 0 , t > 0 with 1 ∕ s + 1 ∕ t = 1 ∕ p , and let X be a random variable. Let { a n i , 1 ≤ i ≤ n , n ≥ 1 } be an array of constants satisfying sup n ≥ 1 n − 1 ∑ i = 1 n ∣ a n i ∣ s < ∞ . Then, for any τ > max { s , ( r − 1 ) t } , we have that

∑ n = 1 ∞ n r − 2 − τ ∕ p ∑ i = 1 n E ∣ a n i X ∣ τ I ( ∣ a n i X ∣ ≤ n 1 ∕ p ) ≤ C E ∣ X ∣ ( r − 1 ) t , i f s < r p , C E ∣ X ∣ ( r − 1 ) t log ( 1 + ∣ X ∣ ) , i f s = r p , C E ∣ X ∣ r p , i f s > r p .

4 Proofs of the main results

Proof of Theorem 2.1

Without loss of generality, we may assume that ∑ i = 1 n ∣ a n i ∣ s ≤ n . Hence, it follows from Hölder’s inequality that ∑ i = 1 n ∣ a n i ∣ q ≤ n for any 0 < q < s . Denote for fixed n ≥ 1 and 1 ≤ i ≤ n that

X n i = a n i X i I ( ∣ a n i X i ∣ ≤ n 1 ∕ p ) .

Since { X n , n ≥ 1 } is defined on the σ -algebras, { X n i , 1 ≤ i ≤ n , n ≥ 1 } is still an array of ( α , β ) -mixing random variables according to Definition 1.1. It is easy to obtain by E X = 0 , E ∣ X ∣ p < ∞ , ∣ a n i ∣ ≤ n 1 ∕ s and the dominated convergence theorem that

n − 1 ∕ p ∑ i = 1 n E X n i = n − 1 ∕ p ∑ i = 1 n E ( a n i X i − X n i ) ≤ n − 1 ∕ p ∑ i = 1 n E ∣ a n i X ∣ I ( ∣ a n i X ∣ > n 1 ∕ p ) ≤ n − 1 ∑ i = 1 n a n i p E ∣ X ∣ p I ( ∣ a n i X ∣ > n 1 ∕ p ) ≤ E ∣ X ∣ p I ( ∣ X ∣ > n 1 ∕ t ) → 0 , as n → ∞ .

Therefore, for any ε > 0 and all n large enough,

(4.1) ∑ i = 1 n E X n i ≤ ε n 1 / p / 2 .

Hence, we can easily obtain by (4.1) that

∑ n = 1 ∞ n r − 2 P ∑ i = 1 n a n i X i > ε n 1 ∕ p ≤ ∑ n = 1 ∞ n r − 2 P ∑ i = 1 n X n i > ε n 1 ∕ p + ∑ n = 1 ∞ n r − 2 P ( ⋃ i = 1 n { ∣ a n i X i ∣ > n 1 ∕ p } ) ≤ C ∑ n = 1 ∞ n r − 2 P ∑ i = 1 n ( X n i − E X n i ) > ε n 1 ∕ p / 2 + ∑ n = 1 ∞ n r − 2 ∑ i = 1 n P ( ∣ a n i X ∣ > n 1 ∕ p ) ≕ I 1 + I 2 .

By Lemma 3.3, we have I 2 < ∞ . Now we will show that I 1 < ∞ . Set u ∈ ( p , min { 2 , s , r p } ) and τ > max { 2 , s , ( r − 1 ) t , 2 p ( r − 1 ) ∕ ( u − p ) } . It is easy to obtain by Lemma 3.1 and the definition of X n i that

I 1 ≤ C ∑ n = 1 ∞ n r − 2 − τ ∕ p E ∑ i = 1 n ( X n i − E X n i ) τ ≤ C ∑ n = 1 ∞ n r − 2 − τ ∕ p ∑ i = 1 n E ∣ X n i ∣ τ + ∑ i = 1 n E X n i 2 τ ∕ 2 ≤ C ∑ n = 1 ∞ n r − 2 − τ ∕ p ∑ i = 1 n E ∣ a n i X ∣ τ I ( ∣ a n i X ∣ ≤ n 1 ∕ p ) + C ∑ n = 1 ∞ n r − 2 − τ ∕ p ∑ i = 1 n E X n i 2 τ ∕ 2 ≕ I 11 + I 12 .

Noting that τ > max { s , ( r − 1 ) t } , we can obtain that I 11 < ∞ by Lemma 3.4. On the other hand, noting that ∣ X n i ∣ ≤ n 1 ∕ p , ∣ X n i ∣ ≤ ∣ a n i X i ∣ , E ∣ X ∣ u < ∞ and τ > 2 p ( r − 1 ) ∕ ( u − p ) implies r − 2 + τ ( 1 − u ∕ p ) ∕ 2 < − 1 , we derive that

I 12 ≤ C ∑ n = 1 ∞ n r − 2 − τ ∕ p n ( 2 − u ) ∕ p ∑ i = 1 n E ∣ a n i X i ∣ u τ ∕ 2 = C ∑ n = 1 ∞ n r − 2 − τ ∕ p n ( 2 − u ) ∕ p ∑ i = 1 n a n i u E ∣ X ∣ u τ ∕ 2 ≤ C ∑ n = 1 ∞ n r − 2 + τ ( 1 − u ∕ p ) ∕ 2 < ∞ .

The proof is completed.□

Proof of Theorem 2.2

We first show that

(4.2) ∑ n = 1 ∞ n − 1 P ∑ i = 1 n a n i X i > ε n < ∞ .

In view of the proof of Theorem 2.1, we only need to deal with I 1 < ∞ for r = p = 1 . Recall that we have proved in the Proof of Theorem 2.1 that ∑ i = 1 n ∣ a n i ∣ q ≤ n for all 0 < q < s . Therefore, we may assume without loss of generality that 1 < s < 2 . By taking τ = 2 , we derive from Lemma 3.1 that

I 1 ≤ C ∑ n = 1 ∞ n − 3 E ∑ i = 1 n ( X n i − E X n i ) 2 ≤ C ∑ n = 1 ∞ n − 3 ∑ i = 1 n E ∣ X n i ∣ 2 ≤ C ∑ n = 1 ∞ n − 3 ∑ i = 1 n E ∣ a n i X ∣ 2 I ( ∣ a n i X ∣ ≤ n ) ,

To prove (2.3), it suffices to show

(4.3) ∑ n = 1 ∞ n − 1 P max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i + − E X i + ) > ε n < ∞

and

(4.4) ∑ n = 1 ∞ n − 1 P max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i − − E X i − ) > ε n < ∞ .

Note that E X + ≤ E ∣ X ∣ < ∞ . Therefore, for any ε > 0 , there exists a positive constant Δ ε depending only on ε such that

(4.5) n − 1 ∑ i = 1 n a n i E X i + I ( X i + > Δ ε ) ≤ E X + I ( X + > Δ ε ) ≤ ε / 4 .

Denote for i ≥ 1 that

X i ( 1 ) = X i + I ( X i + ≤ Δ ε ) , X i ( 2 ) = X i + − X i ( 1 ) = X i + I ( X i + > Δ ε ) .

By (4.5), we can obtain that

(4.6) max 1 ≤ j ≤ n n − 1 ∑ i = 1 j a n i ( X i ( 2 ) − E X i ( 2 ) ) ≤ n − 1 ∑ i = 1 n a n i X i ( 2 ) + n − 1 ∑ i = 1 n a n i E X i ( 2 ) ≤ n − 1 ∑ i = 1 n a n i ( X i ( 2 ) − E X i ( 2 ) ) + 2 n − 1 ∑ i = 1 n a n i E X i ( 2 ) ≤ n − 1 ∑ i = 1 n a n i ( X i ( 2 ) − E X i ( 2 ) ) + ε / 2 .

Therefore, it can be checked by

max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i + − E X i + ) ≤ max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i ( 1 ) − E X i ( 1 ) ) + max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i ( 2 ) − E X i ( 2 ) )

and (4.6) that

∑ n = 1 ∞ n − 1 P max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i + − E X i + ) > ε n ≤ ∑ n = 1 ∞ n − 1 P max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i ( 1 ) − E X i ( 1 ) ) > ε n / 4 + ∑ n = 1 ∞ n − 1 P max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i ( 2 ) − E X i ( 2 ) ) > 3 ε n / 4 ≤ ∑ n = 1 ∞ n − 1 P max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i ( 1 ) − E X i ( 1 ) ) > ε n / 4 + ∑ n = 1 ∞ n − 1 P ∑ i = 1 n a n i ( X i ( 2 ) − E X i ( 2 ) ) > ε n / 4 ≕ J 1 + J 2 .

Take τ ′ > max { s , 2 } . Noting that ∑ i = 1 n ∣ a n i ∣ s ′ ≤ n s ′ ∕ s for any s ′ > s , we obtain ∑ i = 1 n a n i 2 ≤ n 2 ∕ s + n . Hence, it follows from Markov’s inequality and Lemma 3.2 that

J 1 ≤ C ∑ n = 1 ∞ n − 1 − τ ′ E max 1 ≤ j ≤ n ∑ i = 1 j a n i ( X i ( 1 ) − E X i ( 1 ) ) τ ′ ≤ C ∑ n = 1 ∞ n − 1 − τ ′ log τ ′ n ∑ i = 1 n ∣ a n i ∣ τ ′ E ∣ X i ( 1 ) ∣ τ ′ + ∑ i = 1 n a n i 2 E X i ( 1 ) 2 τ ′ ∕ 2 ≤ C ∑ n = 1 ∞ n − 1 − τ ′ log τ ′ n [ n τ ′ ∕ s + ( n 2 ∕ s + n ) τ ′ ∕ 2 ] ≤ C ∑ n = 1 ∞ n − 1 − τ ′ + τ ′ ∕ s log τ ′ n + C ∑ n = 1 ∞ n − 1 − τ ′ ∕ 2 log τ ′ n < ∞ .

By (4.2), we can obtain that J 2 < ∞ . This completes the proof of (4.3). The proof of (4.4) is completely analogous to that of (4.3). Therefore, the proof is completed.□

Proof of Corollary 2.2

Without loss of generality, we may assume that ∑ i = 1 n ∣ a n i ∣ s ≤ n . For any Θ > 0 and n ≥ 1 , we denote for 1 ≤ i ≤ n that

Y i = X i I ( ∣ X i ∣ ≤ Θ ) , Z i = X i − Y i = X i I ( ∣ X i ∣ > Θ ) .

Nothing that ∣ Y i − E Y i ∣ ≤ 2 Θ , and ∑ i = 1 n ∣ a n i ∣ q ≤ n for any 0 < q < s and ∑ i = 1 n ∣ a n i ∣ s ′ ≤ n s ′ ∕ s for any s ′ > s , we obtain by Markov’s inequality and Lemma 3.1 that for any s ′ > max { 2 , t , s } ,

∑ n = 1 ∞ P ∑ i = 1 n a n i ( Y i − E Y i ) > ε n ≤ C ∑ n = 1 ∞ n − s ′ E ∑ i = 1 n a n i ( Y i − E Y i ) s ′ ≤ C ∑ n = 1 ∞ n − s ′ ∑ i = 1 n ∣ a n i ∣ s ′ E ∣ Y i − E Y i ∣ s ′ + ∑ i = 1 n a n i 2 E ∣ Y i − E Y i ∣ 2 s ′ ∕ 2 ≤ C ∑ n = 1 ∞ n − s ′ ∑ n = 1 n ∣ a n i ∣ s ′ + ∑ i = 1 n a n i 2 s ′ ∕ 2 ≤ C ∑ n = 1 ∞ n − s ′ { n s ′ ∕ s + ( n 2 ∕ s + n ) s ′ ∕ 2 } ≤ C ∑ n = 1 ∞ ( n − s ′ ∕ t + n − s ′ ∕ 2 ) < ∞ ,

which together with Borel-Cantelli lemma yields that

(4.7) n − 1 ∑ i = 1 n a n i ( Y i − E Y i ) → 0 a.s.

On the other hand, it follows from Corollary 2.1 and E ∣ X ∣ t < ∞ that n − 1 ∑ i = 1 n ∣ Z i − E Z i ∣ t → E ∣ Z 1 − E Z 1 ∣ t a.s. Hence, we have that

(4.8) n − 1 ∑ i = 1 n a n i ( Z i − E Z i ) ≤ n − 1 ∑ i = 1 n a n i s 1 ∕ s n − 1 ∑ i = 1 n ∣ Z i − E Z i ∣ t 1 ∕ t ≤ C { E ∣ X ∣ t I ( ∣ X ∣ > Θ ) } 1 ∕ t → 0 a.s.

by letting n → ∞ first and then Θ → ∞ . Noting that E X i = 0 , we complete the proof by (4.7) and (4.8).□

Proof of Theorem 2.3

Sufficiency: Suppose that S n ∕ n → ∞ . In view of (2.6), it suffices to show

(4.9) S n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) ε i → 0 a.s. , n → ∞ ,

(4.10) S n − 1 ∑ i = 1 n ( x i − x ¯ n ) ( ε i − β 0 δ i ) → 0 a.s. , n → ∞ ,

(4.11) S n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) 2 → 0 a.s. , n → ∞ ,

and

(4.12) S n − 1 ∑ i = 1 n ( ξ i − ξ ¯ n ) 2 → 1 a.s. , n → ∞ .

We have by S n ∕ n → ∞ and Corollary 2.1 that

S n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) 2 ≤ n S n 1 n ∑ i = 1 n ( δ i 2 − E δ i 2 ) + n S n δ ¯ n 2 + n S n E δ 1 2 → 0 a.s. , n → ∞ .

So (4.11) has been proved. Similarly, we can also obtain

S n − 1 ∑ i = 1 n ( ε i − ε ¯ n ) 2 → 0 a.s. , n → ∞ ,

which together with Cauchy-Schwartz’s inequality and (4.11) yields that

S n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) ε i = S n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) ( ε i − ε ¯ n ) ≤ S n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) 2 1 ∕ 2 S n − 1 ∑ i = 1 n ( ε i − ε ¯ n ) 2 1 ∕ 2 → 0 a.s. , n → ∞ .

Thus, (4.9) has been proved. To prove (4.10), we take a n i = ( x i − x ¯ n ) n ∕ S n for 1 ≤ i ≤ n , n ≥ 1 . It is easy to check that ∑ i = 1 n a n i 2 = O ( n ) . By applying Corollary 2.2 with s = 2 , we can obtain by ( A 2 ) that

S n − 1 ∑ i = 1 n ( x i − x ¯ n ) ε i = n − 1 ∑ i = 1 n a n i ε i → 0 a.s. , n → ∞

and

(4.13) S n − 1 ∑ i = 1 n ( x i − x ¯ n ) δ i = n − 1 ∑ i = 1 n a n i δ i → 0 a.s. , n → ∞ .

Hence, (4.10) has been obtained. It remains to show (4.12). Note by (4.9) and (4.11) that

S n − 1 ∑ i = 1 n ( ξ i − ξ ¯ n ) 2 = 1 + 2 S n − 1 ∑ i = 1 n ( x i − x ¯ n ) δ i + S n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) 2 → 1 .

Necessity: Suppose that S n ∕ n → ∞ as n → ∞ does not hold. Taking a subsequence if necessary, we can assume that S n ∕ n → c for some 0 ≤ c < ∞ . It follows from ( A 2 ) and Cauchy-Schwartz’s inequality that

E ∣ ε 1 δ 1 ∣ ≤ ( E ε 1 2 ) 1 ∕ 2 ( E δ 1 2 ) 1 ∕ 2 < ∞ .

Therefore, it follows from Corollary 2.1 that

(4.14) n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) ε i = n − 1 ∑ i = 1 n ε i δ i − δ ¯ n ε ¯ n → E ε 1 δ 1 a.s.

and

(4.15) n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) 2 = n − 1 ∑ i = 1 n δ i 2 − δ ¯ n 2 → E δ 1 2 a.s.

Set a n i = x i − x ¯ n for 1 ≤ i ≤ n , n ≥ 1 . It is easy to check that ∑ i = 1 n a n i 2 = S n = O ( n ) . By applying Corollary 2.2 with s = 2 , we have that

(4.16) n − 1 ∑ i = 1 n ( x i − x ¯ n ) ε i = n − 1 ∑ i = 1 n a n i ε i → 0 a.s. , n → ∞ ,

and

(4.17) n − 1 ∑ i = 1 n ( x i − x ¯ n ) δ i = n − 1 ∑ i = 1 n a n i δ i → 0 a.s. , n → ∞ .

Moreover, by (4.15) and (4.17), we obtain that

n − 1 ∑ i = 1 n ( ξ i − ξ ¯ n ) 2 = S n n + 2 n ∑ i = 1 n ( x i − x ¯ n ) δ i + n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) 2 → c + E δ 1 2 ,

which together with (2.6) and (4.14)–(4.17) yields that

β ˆ n − β 0 → E ε 1 δ 1 − β 0 E δ 1 2 c + E δ 1 2 a.s.

This contradicts to β ˆ n → β 0 a.s. since E ε 1 δ 1 − β 0 E δ 1 2 ≠ 0 . Hence, we have to conclude that S n ∕ n → ∞ . The proof is completed.□

Proof of Theorem 2.4

Sufficiency: Suppose that n x ¯ n ∕ S n * → 0 . In view of (2.7) and ε ¯ n → 0 a.s. as well as δ ¯ n → 0 a.s. by Corollary 2.1, it suffices to show

(4.18) x ¯ n ( β 0 − β ˆ n ) → 0 a.s. , n → ∞

and

(4.19) limsup n → ∞ ∣ β 0 − β ¯ n ∣ < ∞ a.s.

Similar to the proof of (4.11), we also have that as n → ∞ ,

(4.20) x ¯ n S n * ∑ i = 1 n ( δ i − δ ¯ n ) 2 ≤ n ∣ x ¯ n ∣ S n * 1 n ∑ i = 1 n ( δ i 2 − E δ i 2 ) + n ∣ x ¯ n ∣ S n * δ ¯ n 2 + n ∣ x ¯ n ∣ S n * ⋅ 1 n ∑ i = 1 n E δ i 2 → 0 a.s.

and

(4.21) limsup n → ∞ 1 S n * ∑ i = 1 n ( δ i − δ ¯ n ) 2 ≤ limsup n → ∞ n S n * ⋅ 1 n ∑ i = 1 n E δ i 2 ≤ E δ 1 2 a.s.

Similar to the proofs of (4.9) and (4.15), we have by n x ¯ n ∕ S n * → 0 that as n → ∞ ,

(4.22) x ¯ n S n * ∑ i = 1 n ( δ i − δ ¯ n ) ε i ≤ n ∣ x ¯ n ∣ S n * 1 n ∑ i = 1 n ( δ i − δ ¯ n ) 2 1 2 1 n ∑ i = 1 n ( ε i − ε ¯ n ) 2 1 2 → 0 a.s.

and

(4.23) limsup n → ∞ 1 S n * ∑ i = 1 n ( δ i − δ ¯ n ) ε i = limsup n → ∞ n S n * 1 n ∑ i = 1 n ( δ i − δ ¯ n ) ε i ≤ ∣ E ε 1 δ 1 ∣ a.s.

Note that

∑ i = 1 n n x ¯ n ( x i − x ¯ n ) S n * 2 = n 2 x ¯ n S n ( S n * ) 2 = O ( n )

and

∑ i = 1 n n ( x i − x ¯ n ) S n * 2 = n 2 S n ( S n * ) 2 = O ( n ) .

Applying Corollary 2.2 with s = 2 , a n i = n x ¯ n ( x i − x ¯ n ) ∕ S n * and a n i = n ( x i − x ¯ n ) ∕ S n * respectively, we can obtain that

(4.24) x ¯ n S n * ∑ i = 1 n ( x i − x ¯ n ) ( ε i − β 0 δ i ) = 1 n ∑ i = 1 n n x ¯ n ( x i − x ¯ n ) S n * ε i − 1 n ∑ i = 1 n n x ¯ n ( x i − x ¯ n ) S n * β 0 δ i → 0 a.s. , n → ∞

and

(4.25) 1 S n * ∑ i = 1 n ( x i − x ¯ n ) ( ε i − β 0 δ i ) = 1 n ∑ i = 1 n n ( x i − x ¯ n ) S n * ε i − 1 n ∑ i = 1 n n ( x i − x ¯ n ) S n * β 0 δ i → 0 a.s. , n → ∞ .

By virtue of (2.4), we have that

∑ i = 1 n ( ξ i − ξ ¯ n ) 2 = S n + ∑ i = 1 n ( δ i − δ ¯ n ) 2 + 2 ∑ i = 1 n ( x i − x ¯ n ) δ i .

Hence, it follows from the definition of S n * , (4.15) and (4.25) that

(4.26) min ( 1 , E δ 1 2 ) ≤ liminf n → ∞ ∑ i = 1 n ( ξ i − ξ ¯ n ) 2 S n * ≤ limsup n → ∞ ∑ i = 1 n ( ξ i − ξ ¯ n ) 2 S n * ≤ 1 + E δ 1 2 a.s.

Hence, (4.18) follows from (2.6), (4.20), (4.22), (4.24), and (4.26) immediately. On the other hand, we obtain by (2.6), (4.21), (4.23), (4.25), and (4.26) that

(4.27) limsup n → ∞ ∣ β 0 − β ˆ n ∣ ≤ limsup n → ∞ ∑ i = 1 n ( δ i − δ ¯ n ) ε i + ∑ i = 1 n ( x i − x ¯ n ) ( ε i − β 0 δ i ) + ∣ β 0 ∣ ∑ i = 1 n ( δ i − δ ¯ n ) 2 S n * ⋅ S n * ∑ i = 1 n ( ξ i − ξ ¯ n ) 2 ≤ ∣ E ε 1 δ 1 ∣ + ∣ β ∣ E δ 1 2 min ( 1 , E δ 1 2 ) < ∞ a.s. ,

which implies (4.19).

Necessity: Suppose that n x ¯ n ∕ S n * → 0 doesnot hold. By taking a subsequence if necessary, we assume that n x ¯ n ∕ S n * → c 0 for some c 0 ≠ 0 . It follows from (4.14), (4.15), (4.24), and (4.26) that

liminf n → ∞ ∣ x ¯ n ( β 0 − β ˆ n ) ∣ ≥ liminf n → ∞ n ∣ x ¯ n ∣ S n * n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) ε i − β 0 n − 1 ∑ i = 1 n ( δ i − δ ¯ n ) 2 − x ¯ n ∑ i = 1 n ( x i − x ¯ n ) ( ε i − β 0 δ i ) S n * × S n * ∑ i = 1 n ( ξ i − ξ ¯ n ) 2 ≥ ∣ c 0 ( E ε 1 δ 1 − β 0 E δ 1 2 ) ∣ 1 + E δ 1 2 ,

which together with (2.7) and (4.27), ε ¯ n → 0 a.s. and δ ¯ n → 0 a.s. yields that

liminf n → ∞ ∣ θ ˆ n − θ 0 ∣ ≥ ∣ c 0 ( E ε 1 δ 1 − β 0 E δ 1 2 ) ∣ 1 + E δ 1 2 .

This is a contradiction to θ ˆ n → θ 0 a.s. and thus, we have n x ¯ n ∕ S n * → 0 . The proof is completed.□

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 article.

Funding information: This research was funded by the Outstanding Youth Research Project of Anhui Colleges (2022AH030156).
Author contributions: Wenjing Hu: writing original draft; Wei Wang: data curation; Yi Wu: review, supervision, and funding acquisition. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare no conflict of interest.

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Received: 2023-08-23

Revised: 2023-11-22

Accepted: 2024-03-07

Published Online: 2024-05-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2024-0003

Keywords for this article

(α, β)-mixing random variables; complete convergence; strong law of large numbers; simple linear errors-in-variables model

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