Complete Convergence for Weighted Sums of Widely Acceptable Random Variables under Sublinear Expectations
Using different methods than the probability space, under the condition that the Choquet integral exists, we study the complete convergence theorem for weighted sums of widely acceptable random variables under sublinear expectation space. We proved corresponding theorem which was extended to the sublinear expectations’ space from the probability space, and similar results were obtained.
In the study of probability theory and mathematical statistics, limit theory is an important research topic, which is widely used in the financial sector and other fields. However, the establishment of the classical limit theory requires strict conditions for the certainty model, especially in the practice of financial statistics and financial risk measurement, so its limitations are gradually highlighted by many uncertainties. Therefore, Peng [1–3] made an improvement, proposed a sublinear expectations’ space that can model the probability and distribution of uncertainty, and gave the corresponding theoretical system, which has aroused the attention of the majority of scholars. At present, the limit theory has obtained many excellent results under sublinear expectation. For example, in the early research on sublinear expectation, Peng [1–3] extended the central limit theory in the traditional probability space to the sublinear expectation space. Zhang [4–6] continued his research on extended negatively dependent random variables and obtained Kolmogorov’s strong law of large numbers (SLLN) and a series of inequalities under sublinear expectation. Zhang and Chen  obtain the central limit theorem for weighted sums in sublinear expectations’ space. Feng et al.  obtain a complete convergence of the weighted sum of negatively dependent (ND) sequences in the sublinear expectations’ space. Wang and Wu  study on complete convergence and almost sure convergence under the sublinear expectations. Chen  obtains a SLLN for an independent identically distributed sequence in the sublinear expectations space. Liang and Wu  research on complete convergence and complete integral convergence for extended negatively dependent (END) random variables under sublinear expectations.
Complete convergence is one of the most important problems in limit theory research because of the extensive application of weighted sum in statistics, and its properties attract more scholars to study and discuss. In the study of complete convergence in probability space, statistician Hsu and Robbins  first propose the concept of complete convergence in 1947, which aroused the interest of many scholars. So far, complete convergence has been studied very deeply in probability space, for example, Liang and Su  obtain the complete convergence theorem for weighted sums of negatively associated (NA) sequences and discuss its necessity. Sung , based on the exponential inequality, obtains new complete convergence results for weighted sums of independent random variables. Wu [15, 16] proves the complete convergence theorems for ND sequences and arrays of row-wise ND random variables. Lita  explores the property of complete convergence for END random variables. In this article, we establish the complete convergence theorem for weighted sums of widely acceptable (WA) random variables under sublinear expectations. The results have been obtained by Lang et al.  and have been generalized to the sublinear expectation space.
We use the framework and notations of Peng [1–3]. Let be a given measurable space and let be a linear space of real functions defined on such that if , then for each , where denotes the linear space of (local Lipschitz) functions satisfyingfor some depending on is considered as a space of random variables. In this case, we denote .
Definition 1. (see ). A sublinear expectation on is a function satisfying the following properties: for all , we have(a)Monotonicity: if, then (b)Constant preserving: (c)Subadditivity: , whenever is not of the form or (d)Positive homogeneity: Here, . The triple is called a sublinear expectation space.
Given a sublinear expectation , let us denote the conjugate expectation of byFrom the definition, it is easily shown that, for all ,If , then for any . Next, we consider the capacities corresponding to the sublinear expectations. Let . A function is called a capacity ifIt is called subadditive if for all with . In the sublinear space , we denote a pair of capacities bywhere is the complement set of A. By definition of and , it is obvious that is subadditive, andIf , thenIf , thenThis implies Markov inequality: ,from .
Definition 2. (see ). We define the Choquet integrals/expectations bywith being replaced by and , respectively.
Definition 3. (see ).(i) countably subadditive: is called to be countably subadditive if it satisfies(ii) is called to be countably subadditive if
Definition 4 (identical distribution, see ). Let and be two -dimensional random vectors defined, respectively, in a sublinear expectation spaces and . They are called identically distributed ifwhenever the sublinear expectation is finite. A sequence of random variables is said to be identically distributed if, for each and are identically distributed.
Definition 5. (widely acceptable, see ). Let be a sequence of random variables in a sublinear expectation space. The sequence is called widely acceptable (WA) if for , and for all ,where .
It is obvious that if is a sequence of widely acceptable random variables and functions are all nondecreasing (resp. all nonincreasing), then is also a sequence of widely acceptable random variables.
In the following, let be a sequence of random variables in , and . The symbol stands for a generic positive constant which may differ from one place to another. Let denote . denotes that there exists a constant such that for sufficiently large , and denotes an indicator function.
To prove our results, we need the following two lemmas.
Lemma 1 (see ). Let be a sequence of WA random variables in , with for . Then, for all , we have
Lemma 2. Suppose , , for any constant :(i)Then, Especially, when , we have(ii)If , then, for any ,
3. The Main Results and Their Proofs(A.1)Let be a nondecreasing positive function on , , when . And, for some (A.2)There exists a nondecreasing positive function on , such that and for some ,
Theorem 1. Let be a sequence of WA and identically distributed random variables under sublinear expectations. satisfies (A.1) or (A.2), and . When , assume that (A.1) holds andfor some , or (A.2) holds andWhen , assume thatand (A.1) or (A.2) holds. Let be an array of real positive numbers satisfyingThen, for all ,In particular, if , then
Proof. To simplify our proof, we may assume that ; thus,When , it is obvious that by or . When, follows from . Therefore, there exists a positive integer such thatFor widely acceptable random variables , in order to ensure that the truncated random variables are also widely acceptable, truncated functions should belong to and should be nondecreasing. Denote for thatIt is easily checked thatThus, to prove the desired result (23), we only need to show , , and .
Let be a sequence of any random variable, and we all haveFor each , is a sequence of WA, satisfies the conditions of Lemma 1. Let and in Lemma 1; taking and (30), we obtainFor , it follows from the definition of that . By Cr inequality and (22), we obtainFor , we get, by (26), Cr inequality, , and (A.1) or (A.2) holds, such thatThus, we have proved .
Now, we deal with . For , denote for thatThen,For the of condition (A.2), let , , for all , and if , if , and are nonincreasing for any . Then,For , we can get by (36); then,When , we can get by (17). When , we can get by (16). Next, we prove . Obviously,And, it follows from (22) and (27); when , we haveSo, we obtain , when . It follows that . Thus, to estimate , it suffices to show . Next, we consider the following two cases.
Case 1. .
also is a sequence of WA random variables, and satisfies the conditions of Lemma 1. Applying Lemma 1 with , , and (30), taking , we haveFirstly, we prove . For any , by the Cr inequality and (36), we haveThus,For , it follows from (42) and (22) thatWhen , we can get by (16). Next, we prove .
Let be an even function and , such that or all and if and if or . Then,For every , there exists such that ; thus, by (8) and (44) and from which is nonincreasing for any , we obtainIt follows that . For , we consider the following two cases.
Case 3. .
Similar to Case 2, taking and (A.1) or (A.2) hold, we haveSo, when , we have .
Case 4. .
Let and in Lemma 1; taking and (30), we still can obtain (40). And, similar to Case 1, we also get . Next, we discuss the situation of (A.1) or (A.2) holds.
If (A.1) holds, according to (19), we can get . Then,If (A.2) holds, by (42) and Cr inequality, we haveThen, in order to prove , it suffices to show and .
By (36), we can obtain . Since and , we can get and . Combine and (20), and taking , we can obtainAnd, by (36), we also can get . And, combine ; then,So, we have proved .
Next, we prove . From (3), (22), and (27), we haveSo, we can obtain , when . It follows that . Then, (23) holds.
Obviously, also satisfies the conditions of Theorem 1. Considering instead of in (23), we can obtain (24).
In particular, if , then (25) follows from (23), andTogether with (23)–(25) holds. The proof of Theorem 1 is completed.
We give an example satisfying the conditions of Theorem 1 and obtain corresponding results.
Example 1. Let be a sequence of WA and identically distributed random variables with dominating coefficient under sublinear expectations and is countably subadditive. Assume and for some . Then, for all , it will show thatIf , thenObviously, satisfies (A.1). , we have . Furthermore, is countably subadditive, and by , then we have . Therefore, all the conditions of Theorem 1 are satisfied; then, Theorem 1 holds.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (12061028) and Support Program of the Guangxi China Science Foundation (2018GXNSFAA281011).
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