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 -moments for some . 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 -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 and G-normal distribution , where . 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 as the normalizing factor, we can also obtain a bound on the distance between the normalized sum distribution and the corresponding G-normal random variable where . 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 [5–7, 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 [15-18]. Let be a given measurable space. Let be a linear space of real functions defined on Ω such that if then for each where denotes the linear space of local Lipschitz continuous functions φ satisfying for some , depending on φ. contains all where . We also denote as the linear space of bounded Lipschitz continuous functions φ satisfying for some .

Definition 2.1

A functional : is said to be a sub-linear expectation if it satisfies: for ,

Monotonicity: implies .

Constant preserving: , .

Positive homogeneity: , .

Sub-additivity: whenever is well defined.

The triple is called a sub-linear expectation space.

Remark 2.1

The sub-linear expectation satisfies translation invariance: , .

Definition 2.2

([3])

A set function is called a capacity if it satisfies

, .

, , .

Definition 2.3

For a capacity V, a set A is a polar set if . And we say a property holds “quasi-surely” (q.s.) if it holds outside a polar set.

Definition 2.4

A sub-linear expectation is said to be continuous if it satisfies:

continuity from below: implies , where , .

continuity from above: implies , where , .

A capacity is said to be continuous if it satisfies:

continuity from below: implies , where .

continuity from above: implies , where .

The conjugate expectation of sub-linear expectation is defined by Obviously, for all , . A pair of capacities can be induced as follows: , , .

Definition 2.5

([15-18])

(Independence) () is said to be independent of () if, for each test function , whenever the sub-linear expectations are finite.

is said to be a sequence of independent random variables if is independent of for each .

Let be an n-dimensional random variable on a sub-linear expectation space . We define a functional on such that Then can be regarded as the distribution of under and it characterizes the uncertainty of the distribution of .

Definition 2.6

([15-18])

(Identical distribution) Two n-dimensional random variables , on respective sub-linear expectation spaces and are called identically distributed, denoted by , if whenever the sub-linear expectations are finite.

Definition 2.7

([15-18])

A one-dimensional random variable ξ on sub-linear expectation is said to be G-normal distributed, denoted by , if for any the following function defined by is the unique viscosity solution of the following parabolic partial differential equation (PDE) defined on : where , .

Remark 2.2

The G-normal distributed random variable ξ satisfies: , , where and ξ̅ is independent of ξ. This implies .

Next we recall the definition of G-expectation. Let be a space of all -valued continuous paths with , equipped with distance Denote for each and For each given monotonic and sub-linear function , , , let the canonical process be G-Brownian motion on a G-expectation space . That is, where , .

For each , we denote as the completion of under the norm . Then the G-expectation can be continuously extended to . We still denote the extended G-expectation space by .

Proposition 2.1

([4, 11])

There exists a weakly compact set of probability measures on such that where denotes the Borel σ-algebra of Ω̃. We say that represents .

Given a G-expectation space , we can define a pair of capacities: Obviously, by Proposition 2.1, and are continuous from below.

Definition 2.8

A sub-linear expectation is said to be regular if, for each sequence satisfying , we have .

Lemma 2.1

([4, 11])

For any closed sets , it holds that .

G-expectation is regular.

Hu et al. [11] indicated that G-Brownian motion does not converge to any single point in probability under capacity as follows.

Lemma 2.2

Given a G-expectation space , for any fixed , it holds that In particular, the above equation holds for G-normal distribution .

Lemma 2.3

([8])

implies q.s., i.e., .

The following Rosenthal’s inequality under sub-linear expectation was obtained by Zhang [24].

Proposition 2.2

Assume that is a sequence of independent random variables. Denote . Then, for any , we have where is a positive constant depending on p.

Lemma 2.4

Assume that is continuous from below and . Then If we further assume that is continuous, then

Proof

Since is non-decreasing in n, we have If is continuous, by noting that is non-increasing in n, we have Thus . □

Lemma 2.5

Assume that is continuous from below and regular. Let be a sequence of independent random variables with for any and . Then convergence q.s. under capacity and, for any , we have

One can refer to Zhang and Lin [25] for the proof of the convergence of S. Now we prove (2.2). By , taking on both sides of (2.1), we have On the other hand, Note that . By Lemma 2.4 we have Combining the above inequalities, we have  □

Throughout the rest of this paper, let be a sequence of independent random variables on a sub-linear expectation space with , , , . Denote , , and . 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.

Theorem 2.1

Let ξ be G-normal distributed on a G-expectation space with , . Assume that Then, for any ,

For any ,

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 distribution and G-normal distribution where .

Theorem 3.1

Let ξ be G-normal distributed on a G-expectation space with , .

Then, for any fixed and any , , there exist some , , and (a positive constant depending on h) such that

Remark 3.1

By Theorem 3.1 we can derive Theorem 2.1. If (2.3) and (2.4) hold, taking , , and in turn on both sides of (3.1), we can get (2.5).

For any fixed and any , let . By Definition 2.7, we have that V is the unique viscosity solution of the following parabolic PDE: Let , , , , for each . Then we have Since , for any , it holds that and Then So it is sufficient to get the bound of . where and are obtained by Taylor expansion: Since is independent of , we have Similarly, we can also have . It follows that By the interior regularity of V (see Peng [18]), it holds that which implies , , and are uniformly -Hölder continuous in t and α-Hölder continuous in x on . For any and , it holds that By Proposition 2.2, we have So we have . Similarly . Then On the other hand, So we have Since is independent of and is independent of , we have On the other hand, Then For any , we have By Hölder’s inequality under sub-linear expectation, we have Combining (3.7), (3.8), and (3.9), we have 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 and the corresponding G-normal distribution . And it can also be used to derive the CLT for normalizing factor . We only give the theorem and omit the proof.

Theorem 3.2

Let η be G-normal distributed on a G-expectation space with , .

Then, for any fixed and any , , there exist some , , and (a constant depending on h) such that If we further assume that Then, for any ,

For any ,

Central limit theorem for capacity

The following theorem is the CLT for capacity under the Lindeberg condition.

Theorem 4.1

Assume that Then, for any ,

ξ is G-normal distributed on a G-expectation space with , , and

For any ,

For any fixed , define and It is easy to verify that and . It follows that By Theorem 2.1 we have Then Note that which implies By the arbitrariness of and Lemma 2.2, we have which implies 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 . We omit the details here.

Central limit theorem for summability methods

Let be continuous functions on or λ only valued in . Assume that and, for any , where . Denote , , where for any . Assume that is continuous from below and regular, then by Lemma 2.5, is well defined q.s. under capacity .

Theorem 5.1

Given a sub-linear expectation space , is continuous from below and regular. Assume that Then, for any ,

ξ is G-normal distributed on a G-expectation space with , , and

For any ,

Denote Note that . For any , we can choose N sufficiently large such that .

For any , , and , it holds that Then Hence For any , let and It is easy to verify that and . By the proof process of Theorem 4.1, we have By (3.1), for any , we have Note that , we have In addition, by Lemma 2.5 we have Let . We have and So we have By (5.1), (5.2), Lemma 2.1, and Lemma 2.2, letting , , , , and in turn, we have By the arbitrariness of and , we get On the other hand, for any , , and , Then Hence By the same method as before, we have Letting , , , , and in turn, we have By the arbitrariness of , we have Combining (5.5) with (5.4), we obtain Similarly, we can also have This is equivalent to  □