Complete convergence and complete integral convergence for weighted sums of widely acceptable random variables under the sub-linear expectations

: Since the concept of sub-linear expectation space was put forward, it has well supplemented the deficiency of the theoretical part of probability space. In this paper, we establish the complete convergence and complete integration convergence for weighted sums of widely acceptable (abbreviated as WA) random variables under the sub-linear expectations with the different conditions. We extend the complete moment convergence in probability space to sublinear expectation space.


Introduction
The classical limit theorem considers additive probability and additive expectation, which is suitable for the case of model determination, but this assumption of additivity is not feasible in many fields of practice.As a mathematical theory, nonlinear expectation can be analyzed and calculated under the uncertainty of the mathematical model.In its research, sub-linear expectation plays a special role and is the most studied.Peng [1][2][3] put forward the concept of generalization of sublinear expectation space in 2006, which transforms the probability in probability space into the capacity in sublinear expectation space, which enriches the theoretical part of probability space.Then, after Zhang's [4][5][6][7] research in sublinear expectation space, some important inequalities are obtained.These Volume 7, Issue 5, 8430-8448.
inequalities are a powerful tool for us to study sublinear expectation space.In addition, Zhang also studies the law of iterated logarithm and the strong law of large numbers in sublinear expectation space.After further extension, Wu and Jiang [8] obtained the Marcinkiewicz type strong law of numbers and the Chover type iterated logarithm law for more general cases in sublinear expectation space.
In probability space, the complete convergence and complete moment convergence are two very important research parts.The notion of complete convergence was proposed by Hsu and Robbins [9] in 1947.In 1988, Chow [10] introduced the concept of complete moment convergence.The complete moment convergence is stronger than the complete convergence.The complete convergence and complete moment convergence in probability space have been relatively mature.For example, Qiu [11], Wu [12], and Shen [13] respectively obtained the complete convergence and complete moment convergence for independent identically distributed (i.i.d.), negatively associated (NA), extended negatively dependent (END) random variables sequence in probability space.Due to many methods and tools in probability space, sublinear expectation space can not be used, which increases the difficulty of studying sublinear expectation space, but many scholars have done the research, such as Wu [14] pushed the theorem in Wu [12] from probability space to sublinear expectation space.Feng [15] and Liang [16] obtained the complete convergence and complete integral convergence for arrays of row-wise ND and END random variables respectively.Zhong [17] studied the complete convergence and complete integral convergence for the weighted sum of END random variables.Lu [18] obtained more extensive conditions and conclusions than Zhong [17] in sublinear expectation space.The exponential inequality used in this article was proposed by Anna [19] in 2020.In the inequality, it is assumed that the truncated random variable sequence is a WA random variable sequence.Because it was proposed later, there is little research on WA random variable sequence in sublinear expectation space.Hu [20] proved the complete convergence for weighted sums of WA random variables in 2021.
The organizational structure of this paper is as follows.In Section 2, we summarize some basic symbols and concepts, as well as the related properties in sublinear expectation space, and give a preliminary lemma which is helpful to obtain the main results.In Section 3, We deduce [21] from probability space to sublinear expectation space, obtain the corresponding conclusions, and prove the complete convergence and complete integral convergence for the weighted sums of WA random variables in sublinear expectation space.

Preliminaries
We use the framework and notations of Peng [1][2][3].Let ( , )  be a given measurable space and let be a linear space of real functions defined on ( , )

 =
The triple ( , , )  is called a sub-linear expectation space.
Given a sub-linear expectation ˆ, let us denote the conjugate expectation  of ˆ by ( ) From the definition, it is easily shown that for all ,, XY Next, we consider the capacities corresponding to the sub-linear expectations.Let .


A function : for all , AB with .AB  In the sub-linear space ( , , ),  we denote a pair ( ) where c A is the complement set of A .By definition of and , it is obvious that is subadditive, and This implies Markov inequality: .
X and 2 X be two random variables defined severally in sub-linear expectation space Lemma 2.6.[22] Every regularly varying function (with index 0 where L is a slowly varying function. In  ( ) x respectively, we can get that for 0 c  , ( , x respectively, we have ( ) The last one is the exponential inequality for WA random variables, which was can be found in [19].

Results and proof
Next, we give the theorems and proof in this article.

Let  
;1 n Xn  be a sequence of random variables in sub-linear expectation space 2) is regularly varying function with index  for some 0  is an array of real numbers and there exist some and there also exist a random variable implies that for all 0 .
implies that for any 0, Remark.In Theorem 3.2, we extend the complete moment convergence for the weighted sums of random variables in the probability space of article [21] to the complete integral convergence for the weighted sums of WA random variables in sublinear expectation space.
Proof of Theorem 3.1.Since So, without loss of generality, we can assume that 0 It is easily checked that for 0,   ( )  :.
To prove (3.9), it suffices to show 1 I  and 2 .
In the following, we prove that 2 .I  Firstly, we will show that 1 ˆ0, .
2) and Hӧlder inequality, we have for any 0 q   that ( ) For any 0,   by (3.12) and r C inequality, we have 1, qp  by (3.13), (3.14) and ˆ0, i X = we have for all n large enough, which implies that are WA random variables.Without loss of generality, according to We can verify that  satisfy the conditions in Lemma 2.8 with ( ) : .
Then, we prove 2 .H  Firstly, we will show that The truncation that defines Y  as X is as follows p r p r rp , H  we first need to prove 21 .H  By Markov inequality, r C inequality, (3.12), (3.15), (3.17), Lemma 2.7 (ii), q p r  and 1 H , we have that Volume 7, Issue 5, 8430-8448.According to (3.16) and (3.19), we can get .J  Hence, the finishes the proof of Theorem 3.2.In conclusion, we prove the complete convergence and complete integral convergence for weighted sums of WA random variables under the sub-linear expectations.

Conclusions
In this paper, we extend the conclusion in probability space to sublinear expectation space and obtain the complete convergence and complete integral convergence for weighted sums of WA random variables under the sub-linear expectations, which enriches the limit theory research of WA random If we want to prove (3.5), we just need to prove 9) and (3.10), we can get (3.5).The following proves that (3.9) is established.The definition of   15) Volume 7, Issue 5, 8430-8448.

Proof of Theorem 3 . 2 .
proof of Theorem 3.1 is completed.Without loss of generality, we also can assume that 0
22en, we prove22.H  Similar to previous proof, we consider the following two situations.