A general form for precise asymptotics for complete convergence under sublinear expectation

Abstract: Let {Xn, n ≥ 1} be a sequence of independent and identically distributed random variables in a sublinear expectation (Ω,H , Ê) with a capacity V under Ê. In this paper, under some suitable conditions, I show that a general form of precise asymptotics for complete convergence holds under sublinear expectation. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in studies of complete convergence. The results extend some precise asymptotics for complete convergence theorems from the traditional probability space to the sublinear expectation space. The results also generalize the known results obtained by Xu and Cheng [34].


Introduction
Recently, limit theorems for sublinear expectations have raised a large number of issues of interest, because that the sublinear expectation space has advantages of modelling the uncertainty of probability and distribution. 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 [5] etc.). Peng [1,6,7] introduced the reasonable framework of the sublinear expectation of random variables in a general function space by relaxing the linear property of the classical linear expectation to the subadditivity and positive homogeneity. And sublinear expectation is a natural extension of the classical linear expectation. Later on, more and more limit theorems under sublinear expectation space have been established, which generalize the corresponding fundamental, important limit theorems in probability and statistics. Zhang [8][9][10][11] proved the central limit theorem and Donskers invariance principle, the exponential inequalities , Rosenthals inequalities and Lindeberg's central limit theorems for martingale like sequences under sublinear expectation. Chen [12] proved strong laws of large numbers for sublinear expectation. Wu and Jiang [13] obtained a strong law of large numbers and Chovers law of the iterated logarithm under sublinear expectation. Xu and Zhang [14,15] studied three series theorem and the law of logarithm for arrays of random variables under sublinear expectation. Song [16] obtained normal approximation by Stein's method under sublinear expectation. Liu and Zhang [17,18] established the central limit theorem and the law of iterated logarithm for linear processes generated by independent identically distributed random variables under sublinear expectation. For more results about limit theorems under sublinear expectation, the interested reader could refer to the studies of Chen et al. [19], Wu et al. [20], Feng [21], Fang et al. [22], Zhang [23], Kuczmaszewska [24], Feng et al. [25], Guo and Li [26], and references therein.
Let {X, X n , n ≥ 1} be a sequence of identically distributed random variables with EX = 0 and EX 2 < ∞ in a traditional probability space (Ω, F , P) and define the partial sums S n = n i=1 X i for n ≥ 1. Hsu and Robbins [27] introduced the concept of complete convergence, since then there have been extensions in several directions. One of them is to discuss the precise rate and limit value of ∞ n=1 ϕ(n)P{|S n | ≥ εg(n)} as ε ↓ a, a ≥ 0, where ϕ(x) and g(x) are the positive functions defined on [0, ∞). We call ϕ(x) and g(x) weighted function and boundary function. A first result in this direction was Heyde [28], who proved that where EX = 0 and EX 2 < ∞. For analogous results in more general case, see Spȃtaru [29], Gut and Spȃtaru [30,31]. The research in this field are called the precise asymptotics. Recently, some results on precise asymptotics under sublinear expectation have been obtained. Wu [32] established precise asymptotics for complete integral convergence under sublinear expectation. Zhang [33] established the Heyde's theorem under the sublinear expectation. Xu and Cheng [34] obtained the precise asymptotics in the law of the iterated logarithm under sublinear expectations. The purpose of this paper is to establish the general form of precise asymptotics for complete convergence under sublinear expectation. The paper is organized as follows: In Section 2, some basic concepts and related lemmas under sublinear expectation which are used in this paper are given. In Section 3, the main result of this paper is sated. The proofs of main results are presented in Sections 4 and 5. The conclusion part is listed in Section 6.
Throughout the paper, C denotes a positive constant, which may take different values whenever it appears in different expressions, a n ∼ b n stands for lim n→∞ a n b n = 1, [x] denotes the integer part of x, log x = ln{max{e, x}}, log log x = ln ln{max{e e , x}}.

Preliminaries
Let us recall some notations on sublinear expectation space. More detailed information are referred to Peng [1,6,7]. Let (Ω, F ) be a given measurable space. 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 2 , ..., X n ) ∈ H for each ϕ ∈ C l,Lip (R n ) where ϕ ∈ C l,Lip (R n ) denotes the linear space of local Lipschitz continuous functions ϕ satisfying for some c > 0, m ∈ N depending on ϕ. H contains all I A where A ∈ F . I also denote ϕ ∈ C b,Lip (R n ) as the linear space of bounded Lipschitz continuous functions ϕ satisfying for some c > 0.
Next, I introduce the capacities corresponding to the sublinear expectation. (2.1) In addition, a pair (C V , C V ) of the Choquet integrals/expecations denoted by with V being replaced by V and V, respectively. If E is countably subadditive or for all p > 0 (See Lemma 4.5 (iii) of Zhang [9]).
It is obvious that a continuous subadditive capacity V is countably subadditive. Peng [7] introduced the concept of independent and identically distributed (IID) random variable and G-normal distribution under sublinear expectation. The definitions are as follows.
Definition 2.5. (i) (Identical distribution) Let X 1 and X 2 be two n-dimensional random vectors defined respectively in sublinear expectation spaces (Ω 1 , H 1 , E 1 ) and (Ω 2 , H 2 , E 2 ). They are called identically distributed, denoted by X 1 (iii) (IID random variables) A sequence of random variables {X n , n ≥ 1} is said to be independent and identically distributed (IID), if X i d = X 1 and X i+1 is independent to (X 1 , ..., X i ) for each i ≥ 1.
The last lemma obtained by Wu [32] shows the uniform convergence rate of Berry-Esseen ineqality.
Lemma 2.8. Assume that {X n , n ≥ 1} is a sequence of independent and identically distributed random variables with E[ Suppose that E is continuous and set Then ∆ n = sup x≥0 |∆ n (x)| → 0, as n → ∞.

Main results
At first, I give the following assumptions on boundary functions and weighted functions: (A1) Let g(x) be a positive and differentiable function defined on [n 0 , ∞), which is strictly increasing to ∞.
The following are main results.
Remark 4.2. In the following, without loss of generality, one can assume that ϕ(x) is nonincreasing. For the other case, the discussion process is similar to that of Proposition 4.1.
n=n 0 ϕ(n) ∼ −r log ε, then by Lemma 2.8 and Toeplitz's lemma, one can get Thus the proof is completed. Proof. Since n > b(ε) implies εg s (n) > ε 1−rs . Then by the same argument in Proposition 4.1, using L'Hospital's rule and note that r > 1/s, Proof. By the same argument in Proposition 4.1, Lemma 2.7 (Markov's inequality), σ 2 = E[X 2 ] < ∞ and note that r > 1/s > 0, then Proof. At first I discuss the relations between the integral and the series. If ρ(y) is nonincreasing, then ρ(y)V{|ξ| ≥ εg s (y)} is nonincreasing, hence one has If ρ(y) is nondecreasing, then by lim n→∞ ρ(n + 1)/ρ(n) = 1, the proof is similar to that of Proposition 4.1. Thus Proposition 5.1 is obtained by above steps.

Conclusions
In this paper, using the Markov's inequality and uniform convergence rate of Berry-Esseen ineqality, the author establish a general form of precise asymptotics for complete convergence holds under sublinear expectation. The results extend some precise asymptotics for complete convergence theorems from the traditional probability space to the sublinear expectation space. The results also generalize the known results obtained by Xu and Cheng [34]. Recently, the research about statistical probability convergence and its application is a new trend in probability and statistics, one can refer to [35][36][37][38][39][40][41][42] and references therein for details, I will consider the statistical probability convergence and its application under expectation space in future.