Central limit theorems for sub-linear expectation under the Lindeberg condition

In this paper, we investigate the central limit theorems for sub-linear expectation for a sequence of independent random variables without assumption of identical distribution. We first give a bound on the distance between the normalized sum distribution and G-normal distribution which can be used to derive the central limit theorem for sub-linear expectation under the Lindeberg condition. Then we obtain the central limit theorem for capacity under the Lindeberg condition. We also get the central limit theorem for capacity for summability methods under the Lindeberg condition.


Introduction
Peng [15] put forward the theory of sub-linear expectation to describe the probability uncertainties in statistics and economics which are difficult to be handled by classical probability theory. There has been increasing interest in sub-linear expectation (see, for example, [1,2,4,11,18,26]).
The classical central limit theorem (CLT for short) is a fundamental result in probability theory. Peng [16] initiated the CLT for sub-linear expectation for a sequence of i.i.d. random variables with finite (2 + α)-moments for some α > 0. The CLT for sub-linear expectation has gotten considerable development. Hu and Zhang [10] obtained a CLT for capacity. Li and Shi [13] got a CLT for sub-linear expectation without assumption of identical distribution. Hu [9] extended Peng's CLT by weakening the assumptions of test functions. Zhang and Chen [21] derived a weighted CLT for sub-linear expectation. Hu and Zhou [12] presented some multi-dimensional CLTs without assumption of identical distribution. Li [14] proved a CLT for sub-linear expectation for a sequence of m-dependent random variables. Rokhlin [19] gave a CLT under the Lindeberg condition under classical probability with variance uncertainty. Zhang [22] gained a CLT for sub-linear expectation under a moment condition weaker than (2 + α)-moments. Zhang [23] established a martingale CLT and functional CLT for sub-linear expectation under the Lindeberg condition.
The purpose of this paper is to investigate the CLTs for sub-linear expectation for a sequence of independent random variables without assumption of identical distribution. We first give a bound on the distance between the normalized sum distribution E[ϕ( S n B n )] and G-normal distribution E[ϕ(ξ )], where ξ ∼ N ({0}; [σ 2 , 1]). It can be used to derive the CLT for sub-linear expectation under the Lindeberg condition directly, which coincides with the result in Zhang [23]. Different from the classical case, when choosing B n as the normalizing factor, we can also obtain a bound on the distance between the normalized sum distribution E[ϕ( S n B n )] and the corresponding G-normal random variable E[ϕ(η)] where η ∼ N ({0}; [1, σ 2 ]). Secondly, we obtain a CLT for capacity under the Lindeberg condition which extends the CLT for capacity for a sequence of i.i.d. random variables in Hu and Zhang [10]. We also study the CLT for capacity for summability methods under the Lindeberg condition. The regular summability method is an important subject in functional analysis. In recent years it has been found that summability method plays an important role in the study of statistical convergence (see [5][6][7]20]). So it is meaningful to investigate the CLT for capacity for summability methods.
This paper is organized as follows. In Sect. 2, we recall some basic concepts and lemmas related to the main results. In Sect. 3, we give a bound on the distance between the normalized sum distribution and G-normal distribution. In Sect. 4, we prove a CLT for capacity under the Lindeberg condition. In Sect. 5, we show a CLT for capacity for summability methods under the Lindeberg condition.

Basic concepts and lemmas
This paper is studied under the sub-linear expectation framework established by Peng [15][16][17][18]. Let (Ω, F) be a given measurable space. Let H be a linear space of real functions defined on Ω 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 C l,Lip (R n ) denotes the linear space of local Lipschitz continuous functions ϕ satisfying ϕ(x)ϕ(y) ≤ C 1 + |x| m + |y| m |x -y|, ∀x, y ∈ R n , for some C > 0, m ∈ N depending on ϕ. H contains all I A where A ∈ F . We also denote C b,Lip (R n ) as the linear space of bounded Lipschitz continuous functions ϕ satisfying for some C > 0.  (1) continuity from below: (2) continuity from above: 1] is said to be continuous if it satisfies: (1) continuity from below: (2) continuity from above: The conjugate expectation E of sub-linear expectation E is defined by Obviously, for all . A pair of capacities can be induced as follows: whenever the sub-linear expectations are finite. {X n } ∞ n=1 is said to be a sequence of independent random variables if X n+1 is independent of (X 1 , . . . , X n ) for each n ≥ 1.
Let X be an n-dimensional random variable on a sub-linear expectation space (Ω, H, E). We define a functional on C l,Lip (R n ) such that Then F X [·] can be regarded as the distribution of X under E and it characterizes the uncertainty of the distribution of X. Definition 2.6 ( [15][16][17][18]) (Identical distribution) Two n-dimensional random variables X 1 , X 2 on respective sub-linear expectation spaces (Ω 1 , H 1 , E 1 ) and (Ω 2 , H 2 , E 2 ) are called identically distributed, denoted by X 1 , ∀ϕ ∈ C l,Lip R n , whenever the sub-linear expectations are finite.

Proposition 2.1 ([4, 11]) There exists a weakly compact set of probability measures P on
where B( Ω) denotes the Borel σ -algebra of Ω. We say that P represents E.
Given a G-expectation space ( Ω, L 1 G ( Ω), E), we can define a pair of capacities: Obviously, by Proposition 2.1, E[·] and V(·) are continuous from below.
Hu et al. [11] indicated that G-Brownian motion does not converge to any single point in probability under capacity V as follows.

Lemma 2.2 Given a G-expectation space
In particular, the above equation holds for G-normal distribution W 1 .
The following Rosenthal's inequality under sub-linear expectation was obtained by Zhang [24].
where C p is a positive constant depending on p.

Lemma 2.4
Assume that E is continuous from below and lim n→∞ X n = X. Then If we further assume that E is continuous, then Proof Since inf i≥n X i is non-decreasing in n, we have If E is continuous, by noting that sup i≥n X i is non-increasing in n, we have Thus

Lemma 2.5 Assume that E is continuous from below and regular. Let
(2.2) Proof One can refer to Zhang and Lin [25] for the proof of the convergence of S. Now we prove (2.2). By E[X n ] = E[X n ] = 0, taking lim sup n→∞ on both sides of (2.1), we have On the other hand, Note that lim n→∞ S n = S. By Lemma 2.4 we have Combining the above inequalities, we have Throughout the rest of this paper, let {X n } ∞ n=1 be a sequence of independent random variables on a sub-linear expectation space The symbol C presents an arbitrary positive constant and may take different values in different positions.
Zhang [23] obtained the following CLT for sub-linear expectation under the Lindeberg condition as a corollary of the martingale CLT for sub-linear expectation.

The bound on the distance between the normalized sum distribution and G-normal distribution
The following theorem gives a bound on the distance between the normalized sum distri- Then, for any fixed ϕ ∈ C b,Lip (R) and any h > 0, 0 < ε < 1, there exist some 0 < α < 1, C > 0, and C h > 0 (a positive constant depending on h) such that Proof For any fixed ϕ ∈ C b,Lip (R) and any h > 0, let . By Definition 2.7, we have that V is the unique viscosity solution of the following parabolic PDE: Since ϕ ∈ C b,Lip (R), for any 0 < ε < 1, it holds that So it is sufficient to get the bound of |E[V (1, S (n) n )] -V (0, 0)|.
where I (n) i and J (n) i are obtained by Taylor expansion: By the interior regularity of V (see Peng [18]), it holds that which implies ∂ t V , ∂ x V , and ∂ xx V are uniformly α 2 -Hölder continuous in t and α-Hölder continuous in x on [0, 1] × R. For any n ≥ 1 and i ≤ n, it holds that By Proposition 2.2, we have On the other hand, On the other hand, For any 0 < ε < 1, we have By Hölder's inequality under sub-linear expectation, we have (3.9) Combining (3.7), (3.8), and (3.9), we have (3.10) By (3.4), (3.5), and (3.10), it holds that Thus we obtain (3.1).
By a similar method, we can obtain a bound on the distance between the normalized sum distribution E[ϕ( S n B n )] and the corresponding G-normal distribution E[ϕ(η)]. And it can also be used to derive the CLT for normalizing factor B n . We only give the theorem and omit the proof.

Central limit theorem for capacity
The following theorem is the CLT for capacity under the Lindeberg condition. (2) For any ε > 0, Then, for any a ∈ R, Proof For any fixed ε > 0, define It is easy to verify that f , g ∈ C b,Lip (R) and By Theorem 2.1 we have Note that which implies By the arbitrariness of ε > 0 and Lemma 2.2, we have Similarly, we can also obtain That is, Remark 4.1 By a similar method, we can also obtain the CLT for capacity for the normalized sum S n /B n . We omit the details here.

Central limit theorem for summability methods
Let c i (λ) be continuous functions on (0, ∞) or λ only valued in N * . Assume that 0 ≤ c i (λ) ≤ 1 and, for any λ > 0, Assume that E is continuous from below and regular, then by Lemma 2.5, S λ := ∞ i=1 c i (λ)X i is well defined q.s. under capacity V.
(2) For any ε > 0, Then, for any a ∈ R, Note that lim N→∞ B For any a > 0, t > 0, and X, Y ∈ H, it holds that For any η > 0, let It is easy to verify that f , g ∈ C b,Lip (R) and g(x) ≤ I(x ≤ k(a + t)) ≤ f (x). By the proof process of Theorem 4.1, we have By (3.1), for any 0 < ε < 1, we have In addition, by Lemma 2.5 we have So we have On the other hand, for any a > 0, 0 < t < a, and X, Y ∈ H, Then By the same method as before, we have Letting t = (E[|S * λ,N | 2 ]) 1 3 , N → ∞, λ → ∞, ε → 0, and η → 0 in turn, we have By the arbitrariness of h > 0, we have Similarly, we can also have This is equivalent to