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.


Introduction
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. erefore, Peng [1][2][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][2][3] extended the central limit theory in the traditional probability space to the sublinear expectation space. Zhang [4][5][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 [7] obtain the central limit theorem for weighted sums in sublinear expectations' space. Feng et al. [8] obtain a complete convergence of the weighted sum of negatively dependent (ND) sequences in the sublinear expectations' space. Wang and Wu [9] study on complete convergence and almost sure convergence under the sublinear expectations. Chen [10] obtains a SLLN for an independent identically distributed sequence in the sublinear expectations space. Liang and Wu [11] 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 [12] 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 [13] obtain the complete convergence theorem for weighted sums of negatively associated (NA) sequences and discuss its necessity. Sung [14], 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 [17] 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. e results have been obtained by Lang et al. [18] and have been generalized to the sublinear expectation space.

Preliminaries
We use the framework and notations of Peng [1][2][3]. Let (Ω, F) be a given measurable space and let H be a linear space of real functions defined on (Ω, F) such that if X 1 , X 2 , . . . , X n ∈ H, then φ(X 1 , . . . , X n ) ∈ H for each φ ∈ C l,Lip (R n ), where C l,Lip (R n ) denotes the linear space of (local Lipschitz) functions φ satisfying for some c > 0, m ∈ N depending on φ. H is considered as a space of random variables. In this case, we denote X ∈ H.
Definition 1 (see [2]). A sublinear expectation E on H is a function E: H ⟶ R satisfying the following properties: for all X, Y ∈ H, we have e triple (Ω, H, E) is called a sublinear expectation space.
Given a sublinear expectation E, let us denote the conjugate expectation ε of E by From the definition, it is easily shown that, for all X, Y ∈ H,ε X ≤ÊX, If EY � εY, then E(X + aY) � EX + aEY for any a ∈ R. Next, we consider the capacities corresponding to the sublinear expectations.
where A c is the complement set of A. By definition of V and V, it is obvious that V is subadditive, and is implies Markov inequality: ∀X ∈ H, Definition 2 (see [2]). We define the Choquet integrals/ expectations (C V , C V ) by with V being replaced by V and V, respectively.
(ii) V is called to be countably subadditive if Definition 4 (identical distribution, see [2]). Let X 1 and X 2 be two n-dimensional random vectors defined, respectively, in a sublinear expectation spaces (Ω 1 , H 1 , E 1 ) and (Ω 2 , H 2 , E 2 ). ey are called identically distributed if 2 Discrete Dynamics in Nature and Society whenever the sublinear expectation is finite. A sequence X n ; n ≥ 1 of random variables is said to be identically distributed if, for each i ≥ 1, X i and X 1 are identically distributed.
Definition 5 (widely acceptable, see [19]). Let Y n , n ≥ 1 be a sequence of random variables in a sublinear expectation space(Ω, H,Ê). e sequence Y n , n ≥ 1 is called widely acceptable (WA) if for t ≥ 0, and for all n ∈ N, where 0 < g(n) < ∞.
It is obvious that if X n ; n ≥ 1 is a sequence of widely acceptable random variables and functions . , ∈ C l,Lip (R n ) are all nondecreasing (resp. all nonincreasing), then f n (X n ); n ≥ 1 is also a sequence of widely acceptable random variables. In the following, let X n ; n ≥ 1 be a sequence of random variables in (Ω, H,Ê), and S n � n i�1 X i . e symbol c stands for a generic positive constant which may differ from one place to another. Let a x ∼ b x denote lim x⟶∞ a x /b x � 1. a x ≪ b x denotes that there exists a constant c > 0 such that a x ≤ cb x for sufficiently large x, and I(·) denotes an indicator function.
To prove our results, we need the following two lemmas.

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Proof. To simplify our proof, we may assume that max 1≤i≤n a ni ≤ n − 1 ; thus, For widely acceptable random variables X n ; n ≥ 1 , in order to ensure that the truncated random variables are also widely acceptable, truncated functions should belong to C l,Lip and should be nondecreasing. Denote for 1 ≤ i ≤ n that It is easily checked that (29) us, to prove the desired result (23), we only need to show I 1 < ∞, I 2 < ∞, and I 3 < ∞.

(44)
For every n, there exists k such that 2 k− 1 ≤ n < 2 k ; thus, by (8) and (44) and from z(x) which is nonincreasing for any x > 0, we obtain It follows that I 221 < ∞. For I 222 , we consider the following two cases.