{'fraction_non_alphanumeric': 0.06835983859482554, 'fraction_numerical': 0.01618798955613577, 'mean_word_length': 3.675099866844208, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 16, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 19, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A New Central Limit Theorem under Sublinear Expectations

We describe a new framework of a sublinear expectation space and the related notions and results of distributions, independence. A new notion of G-distributions is introduced which generalizes our G-normal-distribution in the sense that mean-uncertainty can be also described. W present our new result of central limit theorem under sublinear expectation. This theorem can be also regarded as a generalization of the law of large number in the case of mean-uncertainty.


Introduction
The law of large numbers (LLN) and central limit theorem (CLT) are long and widely been known as two fundamental results in the theory of probability and statistics. A striking consequence of CLT is that accumulated independent and identically distributed random variables tends to a normal distributed random variable whatever is the original distribution. It is a very useful tool in finance since many typical financial positions are accumulations of a large number of small and independent risk positions. But CLT only holds in cases of model certainty. In this paper we are interested in CLT with mean and varianceuncertainty.
Recently problems of model uncertainties in statistics, measures of risk and superhedging in finance motivated us to introduce, in [13] and [14] (see also [11], [12] and references herein), a new notion of sublinear expectation, called "G-expectation", and the related "G-normal distribution" (see Def. 4.5) from which we were able to define G-Brownian motion as well as the corresponding stochastic calculus. The notion of G-normal distribution plays the same important rule in the theory of sublinear expectation as that of normal distribution in the classic probability theory. It is then natural and interesting to ask if we have the corresponding LLN and CLT under a sublinear expectation and, in particular, if the corresponding limit distribution of the CLT is a G-normal distribution. This paper gives an affirmative answer. We will prove that the accumulated risk positions converge 'in law' to the corresponding G-normal distribution, which is a distribution under sublinear expectation. In a special case where the mean and variance uncertainty becomes zero, the G-normal distribution becomes the classical normal distribution. Technically we introduce a new method to prove a CLT under a sublinear expectation space. This proof of our CLT is short since we borrow a deep interior estimate of fully nonlinear PDE in [5]. The assumptions of our CLT can be still improved.
This paper is organized as follows: in Section 2 we describe the framework of a sublinear expectation space. The basic notions and results of distributions, independence and the related product space of sublinear will be presented in Section 3. In Section 4 we introduce a new notion of G-distributions which generalizes our G-normal-distribution in the sense that mean-uncertainty can be also described. Finally, in Section 5, we present our main result of CLT under sublinear expectation. For reader's convenience we present some basic results of viscosity solutions in the Appendix. This paper is a new and generalized version of [15] in which only variance uncertainty was considered for random variables instead random random vectors. Our new CLT theorem can be applied to the case where both meanuncertainty and variance-uncertainty cannot be negligible. This theorem can be also regarded as a new generalization of LLN. We refer to [9] and [10] for the developments of LLN with non-additive probability measures.

Basic settings
For a given positive integer n we will denote by x, y the scalar product of x, y ∈ R n and by |x| = (x, x) 1/2 the Euclidean norm of x. We denote by S(n) the collection of n × n symmetric matrices and by S + (n) the non negative elements in S(n). We observe that S(n) is an Euclidean space with the scalar product P, Q = tr[P Q].
In this paper we will consider the following type of spaces of sublinear expectations: Let Ω be a given set and let H be a linear space of real functions defined on Ω 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 H is considered as a space of "random variables". In this case we denote X = (X 1 , · · · , X n ) ∈ H n .
Here we use C l.Lip (R n ) in our framework only for some convenience of technicalities. In fact our essential requirement is that H contains all constants and, moreover, X ∈ H implies |X| ∈ H. In general C l.Lip (R n ) can be replaced by the following spaces of functions defined on R n .
• L ∞ (R n ): the space bounded Borel-measurable functions; • C b (R n ): the space of bounded and continuous functions; : the space of bounded and k-time continuously differentiable functions with bounded derivatives of all orders less than or equal to k; • C unif (R n ): the space of bounded and uniformly continuous functions;  (In many situation (c) is also called property of self-domination). The triple (Ω, H,Ê) is called a sublinear expectation space (compare with a probability space (Ω, F , P)). If only (c) and (d) are satisfiedÊ is called a sublinear functional.

Remark 2.3
Just as in the framework of a probability space, a sublinear expectation space can be a completed Banach space under its natural norm · =Ê[|·|] (see [11]- [16]) and by using its natural capacityĉ(·) induced viaÊ [|·|] (see [4] and [3] It is clear that E Q =E. If (a) holds for E, then for each non negative element X ∈ H, for each Example 2.5 For some ϕ ∈ C l.Lip (R), ξ ∈ H, let ϕ(ξ) be a gain value favorable to a banker of a game. The banker can choose among a set of distribution {F (θ, A)} A∈B(R),θ∈Θ of a random variable ξ. In this situation the robust expectation of the risk for a gamblers opposite to the banker is:

Distributions, independence and product spaces
We now consider the notion of the distributions of random variables under sublinear expectations. Let X = (X 1 , · · · , X n ) be a given n-dimensional random vector on a sublinear expectation space (Ω 1 , H 1 ,Ê). We define a functional on C l.Lip (R n ) byF The triple (R n , C l.Lip (R n ),F X [·]) forms a sublinear expectation space.F X is called the distribution of X.
Definition 3.1 Let X 1 and X 2 be two n-dimensional random vectors defined respectively in sublinear expectation spaces (Ω 1 , H 1 ,Ê 1 ) and (Ω 2 , H 2 ,Ê 2 ). They are called identically distributed, denoted by X 1 It is clear that X 1 d = X 2 if and only if their distributions coincide.
Remark 3.2 If the distributionF X of X ∈ H is not a linear expectation, then X is said to have distributional uncertainty. The distribution of X has the following four typical parameters: The subsets [µ,μ] and [σ 2 ,σ 2 ] characterize the mean-uncertainty and the varianceuncertainty of X. The problem of zero-mean uncertainty have been studied in [P3], [P4]. In this paper the mean uncertainty will be in our consideration.
The following simple property is very useful in our sublinear analysis.
Y has no mean uncertainty. Then we havê .
The following notion of independence plays a key role: In a sublinear expectation space (Ω, H,Ê) a random vector Y = (Y 1 , · · · , Y n ), Y i ∈ H is said to be independent to another random vector X = (X 1 , · · · , X m ), Remark 3.5 In the case of linear expectation, this notion of independence is just the classical one. It is important to note that under sublinear expectations the condition "Y is independent to X" does not implies automatically that "X is independent to Y ".
The independence property of two random vectors X, Y involves only the joint distribution of (X, Y ). The following construction tells us how to construct random vectors with given sublinear distributions and with joint independence. Definition 3.7 Let (Ω i , H i ,Ê i ), i = 1, 2 be two sublinear expectation spaces. We denote by and, for each random variable of the above form Z(ω 1 , ω 2 ) = ϕ(X(ω 1 ), Y (ω 2 )), It is easy to check that the triple (Ω 1 × Ω 2 , H 1 × H 2 ,Ê 1 ×Ê 2 ) forms a sunlinear expectation space. We call it the product space of sublinear expectation of (Ω 1 , H 1 ,Ê 1 ) and (Ω 2 , H 2 ,Ê 2 ). In this way we can define the product space of sublinear expectation of any given sublinear expectation spaces The following property is easy to check.
Proposition 3.8 Let X i be n i -dimensional random vectors in sublinear expectation spaces (Ω i , H i ,Ê i ), for i = 1, · · · , n, respectively. We denote Then Y i , i = 1, · · · , n are random variables in the product space of sublinear The situation "Y is independent to X" often appears when Y occurs after X, thus a very robust sublinear expectation should take the information of X into account. We consider the following example: Let Y = ψ(ξ, θ), ψ ∈ C b (R 2 ), where ξ and X are two bounded random variables in a classical probability space (Ω, F , P ) and θ is a completely unknown parameter valued in a given interval [a, b]. We assume that ξ is independent of X under P in the classical sense.
On the space (Ω, H) with H := {ϕ(X, Y ) : ϕ ∈ C l.Lip (R 2 )}, we can define the following three robust sublinear expectations: But it is seen that only under the sublinear expectation E 3 that Y is independent to X. Remark 3.9 It is possible that the above parameter θ is in fact a function of X and ξ: converges.

G-distributed random variables
Given a pair of d-dimensional random vectors (X, Y ) in a sublinear expectation space (Ω, H,Ê), we can define a function It is easy to check that G : G is also a continuous function.
The following property is classic. One can also check it by using Lemma 2.4.
The classical normal distribution can be characterized through the notion of stable distributions introduced by P. Lévy [6] and [7]. The distribution of whereX is an independent copy of X.
The following G-normal distribution plays the same role as normal distributions in the classical probability theory: (3) and continuous in (0, 0). Then there exists a pair of d-dimensional random vectors (X, Y ) in some sublinear expectation space (Ω, H,Ê) satisfying (2) and the following condition: where (X,Ȳ ) is an independent copy of (X, Y ). The distribution of (X, Y ) is uniquely determine by G.

Example 4.3 For the sublinear functionḠ
and the following condition: where Y is an independent copy of Y . In fact we can take To define the corresponding sublinear expectationÊ, we apply Proposition 4.1 to find a subsetΘ ∈ R d such that Then for each ξ ∈ H of the form It is easy to check that the distribution of Y satisfies (5) and (6). It is the so-called worst case distribution with respect to the subset of mean uncertaintȳ Θ. We denote this distribution by U(Θ).

Example 4.4 For the sublinear and monotone functionĜ
and the following condition: whereX is an independent copy of X. In particular, for each components follows that X has no mean uncertainty: On the other hand, by Proposition 4.1 we can find a bounded subsetΘ ∈ S + (d) IfΘ is a singletonΘ = {Q}, then X is a classical zero-mean normal distributed with covariance Q. In generalΘ characterizes the covariance uncertainty of X.
Proposition 4.8 and Corollary 4.9 show that a G-distribution is a uniquely defined sublinear distribution on (R 2d , C l.Lip (R 2d )). We will show that a pair of G-distributed random vectors is characterized, or generated, by the following with Cauchy condition Remark 4. 6 We will use the notion of viscosity solutions to the generating heat equation (12). This notion was introduced by Crandall and Lions. For the existence and uniqueness of solutions and related very rich references we refer to Crandall, Ishii and Lions [2] (see Appendix for the uniqueness). We note that, in the situation where σ 2 > 0, the viscosity solution (12) becomes a classical C 1+ α 2 ,2+α -solution (see [5] and the recent works of [1] and [18]). Readers can understand (12) in the classical meaning.
u is called a viscosity solution of (12) if it is both super and subsolution.
Then we have We also have the estimates: For each T > 0 there exist constants C, k > 0 such that, for all t, s ∈ [0, T ] and x, y ∈ R, and Moreover, u is the unique viscosity solution, continuous in the sense of (14) and (15), of the generating PDE (12).
Proof. Since We then have (14). Let (X,Ȳ ) be an independent copy of (X, Y ). Since (X, Y ) is G-distributed, then We thus obtain (13). From this and (14) it follows that Thus we obtain (15). Now, for a fixed (t, x, y) be such that ψ ≥ u and ψ(t, x, y) = u(t, x, y). By (13) and Taylor's expansion it follows that, for δ ∈ (0, t), From which it is easy to check that Thus u is a viscosity supersolution of (12). Similarly we can prove that u is a viscosity subsolution of (12).
By the above Proposition, both u andū are viscosity solutions of the G-heat equation (12) with Cauchy condition u| t=0 =ū| t=0 = ϕ. It follows from the uniqueness of the viscosity solution that u ≡ū. In particular Thus (X, Y ) d = (X,Ȳ ).
Then v is the unique viscosity solution of the following parabolic PDE Moreover we have v(t, x + y) ≡ u(t, x, y), where u is the solution of the PDE (12) with initial condition u(t, x, y)| t=0 = ψ(x + y).

Proof of Proposition 4.2
We now proceed to prove Proposition 4.2. Let u = u ϕ be the unique viscosity solution of the G-heat equation (12) with u ϕ | t=0 = ϕ, Then we take Ω = R 2d , H = C l.Lip (R 2d ), ω = (x, y) ∈ R 2d . The corresponding sublinear expectation E[·] is defined by, for each ξ ∈ H of the form ξ(ω) = (ϕ(x, y)) (x,y)∈R 2d ∈ C l.Lip (R 2d ), (1, 0). The monotonicity and sub-linearity of u ϕ with respect to ϕ are known in the theory of viscosity solution. For reader's convenience we provide a new and simple proof in the Appendix (see Corollary 6.4 and Corollary 6.5).
It remains to check that the functional E[·] : C l.Lip (R 2d ) → R forms a sublinear expectation, i.e., (a)-(d) of Definition 2.2 are satisfied. Indeed, (a) is simply the consequence of comparison theorem, or the maximum principle of viscosity solution (see [CIL], the prove of this comparison theorem as well as the sub-additivity (c) are given in the Appendix of [P6]). It is also easy to check that, when ϕ ≡ c, the unique solution of (12) is also u ≡ c; hence (b) holds true. (d) also holds since u λϕ = λu ϕ , λ ≥ 0. The proof is complete.

Central Limit Theorem
We assume furthermore that, converges in law to ξ + ζ: for all functions ϕ ∈ C(R d ) satisfying a polynomial growth condition, where (ξ, ζ) is a pair of G-distributed random vectors and where the sublinear function Yi n converges in law to U(Θ), where the subsetΘ ⊂ R d is defined in (7) forḠ(p) = G(p, 0), p ∈ R d . If we take in particular ϕ(y) = dΘ(y) = inf{|x − y| : x ∈Θ}, then by (8) we have the following generalized law of large number:

Remark 5.3 If Y i has no mean-uncertainty, or in other words,Θ is a singleton:
To our knowledge, the law of large numbers with non-additive probability measures have been investigated with a quite different framework and approach from ours (see [9], [10]).
To prove this theorem we first give Lemma 5. 4 We assume the same condition as Theorem 5.1. We assume furthermore that there exists β > 0 such that, for each A,Ā ∈ S(d) with A ≥Ā, we haveÊ Then (17) holds.
We set δ = 1 n and S 0 = 0. Then with, by Taylor's expansion, We now prove thatÊ[ n−1 i=0 J i δ ] = 0. For the 3rd term of J i δ we have: For the second term, we have, from the definition of the function G, We then combine the above two equalities with ∂ t V + G(DV, D 2 V ) = 0 as well as the independence of ( Thus (22) can be rewritten as But since both ∂ t V and D 2 V are uniformly α-hölder continuous in x and α 2hölder continuous in t on [0, 1] × R, we then have It follows thatÊ As n → ∞ we have On the other hand, for each t, t ′ ∈ [0, 1 + h] and x ∈ R d , we have , and, by (23), It follows form (21) and (23) that Since h can be arbitrarily small we thus have We now give Proof of Theorem 5.1. For the case when the uniform Elliptic condition 19 does not hold, we first follow an idea of Song [17] to introduce a perturbation to prove the above convergence for ϕ ∈ C b.Lip (R d ). According to Definition 3.7 and Proposition 3.8 we can construct a sublinear expectation space (Ω,H,Ē) and a sequence of three random vectors and, moreover, We then use the following perturbation X ε i = X i + εη i for a fixed ε > 0. It is seen that the sequence {(X ε i ,Ȳ i )} ∞ i=1 satisfies all conditions in the above CLT, in particular Thus it is strictly elliptic. We then can apply Lemma 5.4 tō where (ξ, ζ) is G-distributes and Sine ε can be arbitrarily small, it follows that On the other hand, it is easy to check that sup nÊ [|S n |] + E[|ξ + ζ|] < ∞. We then can apply the following lemma to prove that the above convergence holds for the case where ϕ in C(R d ) with a polynomial growth condition. The proof is complete.
Lemma 5.5 Let (Ω,Ĥ,Ê) and ( Ω, H, E) be two sublinear expectation space and let ζ ∈Ĥ and ζ n ∈ H, n = 1, 2, · · · , be given. We assume that, for a given p ≥ 1 Proof. We first prove that the above convergence holds for ϕ ∈ C b (R d ) with a compact support. In this case, for each ε > 0, we can find aφ The convergence must hold since ε can be arbitrarily small. Now let ϕ be an arbitrary C b (R n )-function. For each N > 0 we can find ϕ 1 , ϕ 2 ∈ C b (R d ) such that ϕ = ϕ 1 + ϕ 2 where ϕ 1 has a compact support and ϕ 2 (x) = 0 for |x| ≤ N , and |ϕ 2 (x)| ≤ |ϕ(x)| for all x. It is clear that 6 Appendix: some basic results of viscosity solutions We will use the following well-known result in viscosity solution theory (see Theorem 8.3 of Crandall Ishii and Lions [2]).
Let ϕ be defined on an open neighborhood of (0, T ) × Q 1 × · · · × Q k and such that (t, x 1 , · · · , x k ) is once continuously differentiable in t and twice continuously differentiable in (x 1 , · · · , x k ) ∈ Q 1 × · · · × Q k . Suppose thatt ∈ (0, T ),x i ∈ Q i for i = 1, · · · , k and for t ∈ (0, T ) and x i ∈ Q i . Assume, moreover that there is an r > 0 such that for every M > 0 there is a C such that for i = 1, · · · , k, Then for each ε > 0, there are X i ∈ S(N i ) such that (i) (b i , D xi ϕ(t,x 1 , · · · ,x k ), X i ) ∈ P 2,+ u i (t,x i ), i = 1, · · · , k; (ii) Observe that the above conditions (24) will be guaranteed by having u i be subsolutions of parabolic equations given in the following two theorems, which is an improved version of the one in the Appendix of [16].

Then a similar domination also holds for the solutions: If the sum of initial values
Proof. We first observe that forδ > 0 and for each 1 ≤ i ≤ k, the functions defined byũ i := u i −δ/(T − t) is a subsolution of: whereG i (t, x, v, p, X) := G i (t, x, v +δ/(T − t), p, X). It is easy to check that the functionsG i satisfy the same conditions as G i . Since k i=1 β i u i ≤ 0 follows from k i=2 β iũi ≤ 0 in the limitδ ↓ 0, it suffices to prove the theorem under the additional assumptions To prove the theorem, we assume to the contrary that We will apply Theorem 6.1 for x = (x 1 , · · · , x k ) ∈ R k×N and For each large α > 0 the maximum of w − ϕ α achieved at some (t α , x α ) inside a compact subset of [0, T ) × R k×N . Indeed, since thus t α must be inside an interval [0, T 0 ], T 0 < T and x α must be inside a compact set {x ∈ ×R k×N : sup t∈[0,T0] w(t, x) ≥ m0 2 }. We can check that (see where (t,x) is a limit point of (t α , x α 1 ). Since u i ∈ USC, for sufficiently large α, we have We know thatt > 0 and thus t α must be strictly positive for large α. It follows from Theorem 6.1 that, for each ε > 0 there exists b α i ∈ R, X i ∈ S(N ) such that and such that where A = D 2 ϕ α (x α ) ∈ S(kN ) is explicitly given by The second inequality of (30) implies Thus k i=1 β i p α i = 0. This with (29) and (27) it follows that By (28)-(i) we also have lim α→∞ |p α i | · |x α i − x α 1 | → 0. This, together with the the domination condition (26) of G i , implies The right side tends to zero as α → ∞, which induces a contradiction. The proof is complete. (ii) Lipschitz condition:

Theorem 6.3 (Domination Theorem) Let polynomial growth functions
(iii) domination condition for G i : for fixed constants β i > 0, i = 1, · · · , k, Then the following domination holds: If Proof. We set ξ(x) := (1 + |x| 2 ) l/2 and where l is chosen large enough so that k i=1 |ũ i (t, x)| → 0 uniformly. From condition (i) it is easy to check that for each i = 1, · · · ,ũ i is a subsolution of whereG Here Since η and κ are uniformly bounded, one can choose a fixed but large enough λ > 0 such thatG i (x, v, p, X) satisfies all conditions of G i , i = 1, · · · , k in Theorem 6.2. The proof is complete by directly applying this theorem. We have the following Corollaries which are basic in this paper: Corollary 6.4 (Comparison Theorem) Let F 1 , F 2 : R N × S(N ) → R be given functions satisfying similar conditions (i) and (ii) of Theorem 6.3. We also assume that, for each p, q ∈ R N and X, Y ∈ S(N ) such that X ≥ Y , we have F 1 (p, X) ≥ F 2 (p, Y ).