Precise Asymptotics for Complete Integral Convergence under Sublinear Expectations

The aim of this paper is to study and establish the precise asymptotics for complete integral convergence theorems under a sublinear expectation space. As applications, the precise asymptotics for p ( 0 ≤ p ≤ 2 ) order complete integral convergence theorems have been generalized to the sublinear expectation space context. We extend some precise asymptotics for complete moment convergence theorems from the traditional probability space to the sublinear expectation space. Our results generalize corresponding results obtained by Liu and Lin (2006). There is no report on the precise asymptotics under sublinear expectation, and we provide the method to study this subject.


Introduction
e sublinear expectation space has advantages of modelling the uncertainty of probability and distribution. erefore, limit theorems for sublinear expectations have raised a large number of issues of interest recently. Limit theorems are important research topics in probability and statistics. ey were widely used in finance and other fields. Classical limit theorems only hold in the case of model certainty. However, in practice, such model certainty assumption is not realistic in many areas of applications because the uncertainty phenomena cannot be modeled using model certainty. Motivated by modelling uncertainty in practice, Peng [1] introduced a new notion of sublinear expectation. As an alternative to the traditional probability/expectation, capacity/sublinear expectation has been studied in many fields such as statistics, finance, economics, and measures of risk (see Denis and Martini [2]; Gilboa [3]; Marinacci [4]; Peng [1,[5][6][7], etc.). e general framework of the sublinear expectation in a general function space was introduced by Peng [1,7,8], and sublinear expectation is a natural extension of the classical linear expectation.
Because the sublinear expectation provides a very flexible framework to model sublinear probability problems, the limit theorems of the sublinear expectation have received more and more attention and research recently. A series of useful results have been established. Peng [1,7,8] constructed the basic framework, basic properties, and central limit theorem under sublinear expectations, Zhang [9][10][11] established the exponential inequalities, Rosenthal's inequalities, strong law of large numbers, and law of iterated logarithm, Hu [12], Chen [13], and Wu and Jiang [14] studied strong law of large numbers, Wu et al. [15] studied the asymptotic approximation of inverse moment, Xi et al. [16] and Lin and Feng [17] studied complete convergence, and so on. In general, extending the limit properties of conventional probability space to the cases of sublinear expectation is highly desirable and of considerably significance in the theory and application. Because sublinear expectation and capacity is not additive, many powerful tools and common methods for linear expectations and probabilities are no longer valid, so that the study of the limit theorems under sublinear expectation becomes much more complex and difficult.
Since the concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [18], there have been extensions in several directions. One of them is to discuss the precise rate, which is more exact than complete convergence. Precise asymptotics for complete convergence and complete moment convergence is one of the most important problems in probability theory. Many related results have been obtained in the probabilistic space. eir recent results can be found in the studies of Heyde [19]; Liu and Lin [20]; Zhao [21]; Li [22]; Zhou [23]; Gut and Steinebach [24]; He and Xie [25]; Wang et al. [26][27][28][29]; Spaataru [30]; and Kong and Dai [31]. However, in sublinear expectations, due to the uncertainty of expectation and capacity, the precise asymptotics is essentially different from the ordinary probability space.
e study of precise asymptotics of complete convergence and complete integral convergence for sublinear expectations is much more complex and difficult. e precise asymptotics theorems under sublinear expectation have not been reported. e purpose of this paper is to establish the precise asymptotics theorems for p (0 ≤ p ≤ 2) order complete integral convergence for independent and identically distributed random variables under sublinear expectation. As a result, the corresponding results obtained by Liu and Lin [20] have been generalized to the sublinear expectation space context.
In the next section, we summarize some basic notations and concepts and related properties under the sublinear expectations.

Preliminaries
We use the framework and notations of Peng [8]. 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 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. 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.
In the sublinear space (Ω, H, E), we denote a pair (V, ]) of capacities by where A c is the complement set of A. By definition of V and ], it is obvious that V is subadditive, and is implies Markov inequality: ∀X ∈ H, from I(|X| ≥ x) ≤ |X| p /x p ∈ H. By Lemma 4.1 in Zhang [10]; we have Holder inequality: ∀X, Y ∈ H, p, q > 1 sat- And particularly, we have Jensen inequality: ∀X ∈ H, Also, we define the Choquet integrals/expectations with V being replaced by V and ], respectively.
(i) Identical distribution: let X 1 and X 2 be two n-dimensional random vectors defined, respectively, in sublinear expectation spaces (Ω 1 , Mathematical Problems in Engineering whenever the subexpectations are 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. (ii) Independence: in a sublinear expectation space (iii) Independent random variables: a sequence of random variables X n ; n ≥ 1 is said to be independent, if X i+1 is independent of (X 1 , . . . , X i ) for each i ≥ 1.
In the following, let X n ; n ≥ 1 be a sequence of random variables in (Ω, H, E) 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 denote that there exists a constant c > 0 such that a x ≤ cb x for sufficiently large x, and I(·) denote an indicator function.
To prove our results, we need the following four lemmas.
where B n � n k�1 EX 2 k .

Lemma 3 (Theorem 3.3 and Remark 3.4 in Peng
In particular, if σ � σ, then Lemma 3 becomes a classical central limit theorem. Lemma 4 (Lemma 3 in Chen and Hu [32]). Suppose that Let P be a probability measure and φ be a bounded continuous function on R. If B t t≥0 is a Brownian motion under P, then where ) under E, then for each convex function φ, but if φ is a concave function, the above σ must be replaced by σ. If σ � σ � σ, then N(0, [σ 2 , σ 2 ]) � N(0, σ 2 ) which is a classical normal distribution.
In particular, notice that φ(x) � |x| p , p ≥ 1, is a convex function; taking φ(x) � |x| p , p ≥ 1, in (23), we get (24) implies that Remark 2. Lemma 5 is a powerful tool for studying the uniform convergence of the central limit theorem under sublinear expectations, which plays a key role in proving the theorems in this paper.
Proof of Lemma 5. If σ � σ, then Lemma 3 is a classical central limit theorem. In the classical probability, (26) follows from the central limit theorem and an important fact that P(|ξ| ≥ x) is a continuous function of x. erefore, we only need to prove the situation σ < σ.

Mathematical Problems in Engineering
Write Obviously, 0 ≤ F n (x) and F(x) ≤ 1; F n (x) and F(x) are nonincreasing functions on [0, +∞). us, for any x 0 > 0, the limit lim x⟶x 0 F(x) exists. Actually, taking x n ↑x 0 and x n ′ ↓x 0 , by continuity of E, we have Hence, F(x) is continuous for 0 ≤ x < ∞. As well as erefore, let ϵ be an arbitrary positive number; there exist points 0 < x 1 < x 2 , . . . , x m < ∞ satisfying the conditions Furthermore, by (19) and Remark 1, there exists a number n 0 such that for n > n 0 and we have If x k ≤ x < x k+1 (k � 1, . . . , m − 1), then for n > n 0 , we get If 0 < x < x 1 , then for n > n 0 , If x ≥ x m , then for n > n 0 , us, |F n (x) − F(x)| < ε for all x and n > n 0 . at is, (31) holds.

Main Results and Proofs
Our results are stated as follows. Theorem 1. Let X, X n ; n ≥ 1 be a sequence of independent and identically distributed random variables in (Ω, H, E). We assume that E is continuous and here and later, ξ ∼ N(0, [σ 2 , σ 2 ]) under E.

Theorem 2. Under the conditions of eorem 1, for
For p � 2, we have the following theorem.
Theorem 3. Let X, X n ; n ≥ 1 be a sequence of independent and identically distributed random variables in (Ω, H, E). We assume that E is continuous and en, Remark 3. eorems 1-3 extend the corresponding results obtained by Liu and Lin [20] from the probability space to sublinear expectation space.
Since −X, −X i also satisfies the conditions of eorem 1, we replace the X, X i with the −X, −X i in the upper form: (57) erefore, (58) More generally, for any x > 0 and n > Mε − 2 , we have is implies from Markov's inequality and (24) that Let ε ⟶ 0 first; then, let M ⟶ ∞; we get lim ε⟶0 from (44) and (15). From this, combining with (53) and (54), (51) is established. is completes the proof of eorem 1.

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Proof of eorem 2. Since C V (|S n | p I(|S n | ≥ εn)) � V(|S n | ≥ εn) when p � 0, so by eorem 1, we only discuss the case 0 < p < 2. Note that Hence, from eorem 1, in order to establish (46), it suffices to prove that Mathematical Problems in Engineering 7 Let M ≥ 12. Note that Hence, in order to establish (63), it suffices to prove that We first prove (65); let y ≔ x/ � n √ , then from (25). Now, we prove (66). Let b n � ( where , one can easily obtain that