Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables under sub-linear expectation

In this paper, we study the complete convergence and complete moment convergence for weighted sums of extended negatively dependent (END) random variables under sub-linear expectations space with the condition of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\mathbb{V}}[|X|^{p}l(|X|^{1/\alpha})]<\infty$\end{document}CV[|X|pl(|X|1/α)]<∞, further \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{\mathbb {E}}(|X|^{p}l(|X|^{1/\alpha}))\leq C_{\mathbb{V}}[|X|^{p}l(|X|^{1/\alpha })]<\infty$\end{document}Eˆ(|X|pl(|X|1/α))≤CV[|X|pl(|X|1/α)]<∞, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1< p<2$\end{document}1<p<2 (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l(x)>0$\end{document}l(x)>0 is a slow varying and monotone nondecreasing function). As an application, the Baum-Katz type result for weighted sums of extended negatively dependent random variables is established under sub-linear expectations space. The results obtained in the article are the extensions of the complete convergence and complete moment convergence under classical linear expectation space.


Introduction
Additivity has been generally regarded as a fairly natural assumption, so the classical probability theorems have always been considered under additive probabilities and the linear expectations. However, many uncertain phenomena do not satisfy this assumption. So Peng [-] introduced the notions of sub-linear expectations to extend the classical linear expectations. He also established the general theoretical framework of the sublinear expectation space. The theorems of sub-linear expectations are widely used to assess financial riskiness under uncertainty. For complete convergence and complete moment convergence, there are few reports under sub-linear expectations. This paper aims to obtain the complete convergence and complete moment convergence under sub-linear expectation space with the condition of C V [|X| p l(|X| /α )] < ∞, furtherÊ(|X| p l(|X| /α )) ≤ C V [|X| p l(|X| /α )] < ∞,  < p < . In addition, the results and conditions of this paper include a slow varying and monotone nondecreasing function, so the theorems are more generic than the traditional complete convergence. In a word, it is meaningful that this paper extends the complete convergence and complete moment convergence under sublinear expectation.
Sub-linear expectations generate lots of interesting properties which are unlike those in linear expectations, and the issues in sub-linear expectations are more challenging, so lots of scholars have attached importance to them. Numbers of results have been established, for example, Peng [-] gained a weak law of large numbers and a central limit theorem under sub-linear expectation space. Chen [] gained the law of large numbers for independent identically distributed random variables with the condition ofÊ(|X| +α ) < ∞. The powerful tools as the moment inequalities and Kolmogorov's exponential inequalities were established by Zhang [-]. He also obtained the Hartman-Wintner's law of iterated logarithm and Kolmogorov's strong law of large numbers for identically distributed and extended negatively dependent random variables. Wu and Chen [] also researched the law of the iterated logarithm, and Cheng [] studied the strong law of larger number with a general moment condition sup i≥Ê [|X i |ψ(|X i |)] < ∞, and so on. Many powerful inequations and conventional methods for linear expectation and probabilities are no longer valid, the study of limit theorems under sub-linear expectation becomes much more challenging.
The complete convergence has a relatively complete development in probability limit theory. The notion of complete convergence was raised by Hsu  [] researched complete convergence and complete moment convergence for a class of random variables, and so on. In addition, the theorems of this paper are the extensions of the literature [] under sublinear expectation space. And we prove the theorems in this paper with the condition of C V [|X| p l(|X| /α )] < ∞, furtherÊ(|X| p l(|X| /α )) ≤ C V [|X| p l(|X| /α )] < ∞,  < p <  (l(x) >  is a slow varying function).
In the next section, we generally introduce some basic notations and concepts, related properties under sub-linear expectations and preliminary lemmas that are useful to prove the main theorems. In Section , the complete convergence and complete moment convergence under sub-linear expectation space are established. The proofs of these theorems are stated in the last section.

Basic settings
The study of this paper uses the framework and notations which are established by Peng [-]. So, we omit the definitions of sub-linear expectation (Ê), capacity (V, v), countably sub-additive and Choquet integrals/expectations (C V , C v ) and so on.

Definition . (Peng [, ], Zhang [])
(i) (Identical distribution) Assume that a space X  and a space X  are two n-dimensional random vectors defined severally in the sub-linear expectation space (  , H  ,Ê  ) and (  , H  ,Ê  ). They are named identically distributed if whenever the sub-expectations are finite. A sequence {X n , n ≥ } of random variables is named to be identically distributed if, for each i ≥ , X i and X  are identically distributed. (ii) (Extended negatively dependent) A sequence of random variables {X n , n ≥ } is named to be upper (resp. lower) extended negatively dependent if there is some . . , are all nondecreasing (resp. all nonincreasing). They are named extended negatively dependent if they are both upper extended negatively dependent and lower extended negatively dependent.
It is distinct that if {X n , n ≥ } is a sequence of extended independent random variables and f  (x), f  (x), . . . ∈ C l,Lip (R), then {f n (X n ), n ≥ } is also a sequence of extended dependent random variables with K = ; if {X n , n ≥ } is a sequence of upper extended negatively dependent random variables and f  (x), f  (x), . . . ∈ C l,Lip (R) are all nondecreasing (resp. all nonincreasing) functions, then {f n (X n ); n ≥ } is also a sequence of upper (resp. lower) extended negatively dependent random variables. It shall be noted that the extended negative dependence of {X n , n ≥ } underÊ does not imply the extended negative dependence underε.
In the following, let {X n , n ≥ } be a sequence of random variables in ( , H,Ê) and n i= X i = S n . The symbol C is on behalf of a generic positive constant which may differ from one place to another. Let a n b n denote that there exists a constant C >  such that a n ≤ Cb n for sufficiently large n, I(·) denotes an indicator function, a x ∼ b x denotes lim x→∞ a x b x = . Also, let a n ≈ b n denote that there exist constants c  >  and c  >  such that c  a n ≤ b n ≤ c  a n for sufficiently large n.
The following three lemmas are needed in the proofs of our theorems.

Lemma . ([]) l(x) is a slow varying function if and only if
Proof (i) By Lemma ., we can express l(x) as equality (.), and f (u) →  as u → ∞, c(x) → c as x → ∞. Let Z(x) = |x| p l(|x| /α ), Z - (x) be the inverse function of Z(x), l(x) is a slow varying function and for any c > , we have So, (ii) By the proof of (i), we can imply that for any θ > 

Main results
Then, for any ε > ,

Proof
Proof of Theorem . Without loss of generality, we can assume thatÊX i = , when p > .
We just need to prove (.). Because of considering {-X n ; n ≥ } instead of {X n ; n ≥ } in (.), we can obtain (.). Noting that a ni ≥ , without loss of generality, we can assume that For fixed n ≥ , denote for  ≤ i ≤ n that It is easily checked that for ∀ε > , For  < μ < , let g(x) be a decreasing function and g(x) ∈ C l,Lip (R),  ≤ g(x) ≤  for all x and g(x) =  if |x| ≤ μ, g(x) =  if |x| > . Then In order to prove (.), it suffices to show I  < ∞ and I  < ∞. By Lemma .(i) and identically distributed random variables, we can get that In the following, we prove that I  < ∞. First, we prove that For any r > , by the C r inequality and (.), So, by (.) we can imply that By (.), we can imply that and V(|X| > μn α ) ↓, so we get nV(|X| > μn α ) →  as n → ∞. Next, we estimate I  . Let g j (x) ∈ C l,Lip (R), j ≥  such that  ≤ g j (x) ≤  for all x and g j ( For every n, there exists k such that  k- ≤ n <  k , thus by (.), g(x) ↓, and n -α+ ↓ , from α >  p ≥ , we get Noting that by (.), αp > , It follows that I  →  as n → ∞ from the Kronecker lemma and  j(α-) ↑ ∞.
It follows that for all n large enough, which implies that By Definition .(ii), we can know that fixed n ≥ , {a ni (X (n) i -ÊX (n) i ),  ≤ i ≤ n} are still END random variables. Hence, we have by Lemma . (taking x = εn α ) that By (.), we have By Lemma .(i), we can get I  < ∞. Noting that by (.) By p < , we get I  < ∞. Next we estimate I  . By (.), we can imply that Hence, it follows that By I  < ∞ and I  < ∞, we can get I  < ∞.
This finishes the proof of Theorem ..