arXiv:1501.00771v2 [math.PR] 20 Dec 2017 ξ and hn n hntiScrte nttt o iaca Stud Nationa Financial by supported for was work Institute This Securities [email protected]. Zhongtai and China where S esr of measure hoe 1.1 Theorem incomplet for regions confidence convenience: robust readers’ suitably functio construct belief to for short) 3.1) for (CLT theorem limit central a Recently, Introduction 1 one admvrals hs eut r aua xeso o extension natural Epstein are in theorem results measures. limit probability These central additive the variables. generalize we random paper, bounded this In [9]. al. eta ii hoesfrbuddrno aibe ne b under variables random bounded for theorems limit Central . ≤ ∗ ntetermaoe tutr parameter structure above, theorem the In Λ Abstract e words Key colo ahmtc n uniaieEoois Shandon Economics, Quantitative and Mathematics of School c A θ si h etrsense. vector the in is 1 ξ sthe is .,A ..., , sa is A J J J j eetyanwtp fcnrllmttermfrble ucin wa functions belief for theorem limit central of type new a Recently . dmninlnra admvral ihzr enadcovariance and mean zero with variable random normal -dimensional ntefirst the in × are Let eta ii hoe,ble esr,nnadtv measure non-additive measure, belief theorem, limit central . J J ymti n oiiesmdfiiemti ( matrix semidefinite positive and symmetric Λ θ ust of subsets ν n θ ∞ → n ( ∩ n Λ j J xeiet ln h sample the along experiments =1 ∈ { R cov s S J ∞ Ψ . · J θ : ( and √ A n i n ( A , s [ ν ∞ c θ n j n )( N = ) → ( A A J ioi Shi Xiaomin ( measures j j c θ ν c = ) ) )= Λ) ; θ ∈ and − ( A R Ψ i n J 1 aua cec onaino hn N.11401091). (No. China of Foundation Science Natural l e,Sadn nvriy ia,200,P hn.Email: China. PR 250100, Jinan, University, Shandong ies, 1 n ∩ J Then . i P ( =1 P n s A 1 = ∞ ( j I ξ ) )( { nvriyo iac n cnmc,Jnn250014, Jinan Economics, and Finance of University g s ≤ , s − ∞ A i 2 ∗ ∈ ...... ν c j A ( = ) )] θ j , ( } cov A ≤ swsgvni pti ta.[](Theorem [9] al. et Epstein in given was ns j , s efie prior. a fixed be i 1 ) c oes esaeterCTfrthe for CLT their state We models. e θ s , ν h lsia eta ii hoyfor theory limit central classical the f nj ( θ 1 = A 2 ( } ... , A i ) A , j → .,J ..., , ta.[]t comdt general accommodate to [9] al. et ) ). . j )and )) N J ( r h miia frequency empirical the are c Λ) ; ν S θ . arxΛadrelation and Λ matrix safiiesaespace, state finite a is sable ucinon function belief a is ie nEsenet Epstein in given s elief (1.1) (1.2) For more clearly, let Λθ Λ,cn c, then Theorem 1.1 reduced to n ≡ ≡ J ν∞( s∞ : √n[ν(Aj ) Ψ(s∞)(Aj )] c ) NJ (c; Λ). (1.3) ∩j=1{ − ≤ } → Suppose further that J =2, A S, A = S A , then convergence relation (1.1) boils down to 1 ⊂ 2 \ 1 c1 c2 ν∞( + ν(A ) Ψn(s∞)(A ) ν∗(A )+ ) NJ (c; Λ), (1.4) −√n 1 ≤ 1 ≤ 1 √n →

ν(A1)(1 ν(A1)) ν(A1)ν(A2) where c = (c1,c2)′ and Λ =  − −  . ν A ν A ν A ν A  ( 1) ( 2) ( 2)(1 ( 2))   − −  Formula (1.4) shows that the belief measure of Ψn(s∞)(A1) in a two-sided interval will converge to a bivariate normal distribution which is completely different from the classical CLTs under probabilities. To our best knowledge, the above theorem is the first result concerning CLTs for two-sided intervals under belief measures. Formula (1.4) dazzles us because we could hardly imagine the belief measure for a two- sided interval should be approximated by a bivariate normal distribution. Although belief measures, also called infinitely monotone capacities, are the most special non-additive measures (see page 76 in [27]), the above result is a milestone in studying CLTs under non-additive probabilities. It is well known the classical LLNs (laws of large numbers) and CLTs play a significant role in the development of probability theory and its applications. The key in the proof of these theorems lies in the additivity of probabilities. Such additivity assumption is widely used in statistics, economics, decision theory etc, but it is yet not reasonable in many cases. For instance, the famous Allais paradox and Ellsberg paradox (see Allais [1] and Ellsberg [6] respectively) in economic decision theory challenged the additivity assumption a lot. In the seminal paper [3], Choquet proposed the theory of capacities where he removed the additivity assumption. Choquet’s theory has been applied to statistics by Huber and Strassen [13] where Neyman-Pearson lemma for capacities was studied. Also, the Allais paradox and Ellsberg paradox could be explained partially by capacities, or more generally, non-additive probabilities. As the role of classical limit theories played in the probability theory, limit theories under non-additive probabilities are urgent for the more potential general applications.

Given a sequence Xi ∞ of independent and identically distributed (i.i.d. for short) random variables { }i=1 for non-additive probabilities (or capacities), there have been a number of papers related to strong laws of large numbers. We refer to [2],[5],[10],[16],[17],[26]. But to the best of our knowledge, only a few papers studied central limit theorems for non-additive probabilities. Motivated by measuring risk and other financial problems with uncertainty, Peng [18],[19],[22],[23] put forward the notion of i.i.d. random variables in the sublinear expectations space (G-expectation especially), which accommodated volatility uncertainty. G- expectation had many applications in mathematical finance. For instance, Epstein and Ji [7],[8] formulated a model of utility that captures the decision-maker’s concern with ambiguity about both the drift and volatility of the driving process under G framework. And Peng [20],[21] gave a central limit theorem that the accumulated independent and identically distributed random variables converge in law to the G-normal distribution. As for CLT and laws of large numbers under sublinear expectation, we also refer to [12], [14], [15], and [28].

2 But all the above work consider problems from the perspective of sublinear expectations. As we all known, there is a one-to-one correspondence between the set of linear expectations and the set of σ-additive probability measures. But in the sublinear case, this correspondence no longer holds true. Thus CLTs under sublinear expectations neither imply nor are implied by CLTs under non-additive probabilities. For CLTs under non-additive probabilities, Marinacci [17] proves a CLT for a class of capacities that he calls “controlled”. But he only considers CLTs for unidirectional intervals which looks the same as the classical

CLTs as long as the partial sums of Xi ∞ is standardized by proper parameters. As we have mentioned, { }i=1 Theorem 3.1 in Epstein et al. [9] is the first result concerning CLTs for two-sided intervals under belief measures. And it has a nice application in Jovanovic entry game and binary experiments, or more generally, in an environment where the agent does not have enough information to formulate a probability prediction so he can only hold a coarse picture (belief) about the uncertainty. In fact, they construct robust confidence regions for models in which a probabilistic prediction is absent via this central limit theorem. But Theorem 3.1 in [9] considers only Bernoulli random variables which limit the potential application of the CLTs for belief measures. If one were in a belief measure world with general (continuum) state space and general bounded random variables, Theorem 1.1 does not work. So we generalize the CLTs of (1.4) type to accommodate general bounded random variables. But we don’t involve the concrete applications in this paper. And we think it’s still a long way to obtain CLTs under general non-additive probabilities while we hope CLTs under belief measures will give some hints. Extending from Bernoulli random variables to general bounded random variable is interesting and far from trivial. For instance, if random variables X and Y are of Bernoulli type, the expectations EX and E(XY ) are easy to calculate by enumeration. But for general random variables X and Y , the expectations ¯ EX and E(XY ) are not easy to get, especially E(XY ). In the following, random variables Zi and Zi are ∞ ∞ ∞ ∞ Pν Pν 2 Pν ¯ Pν ¯ 2 not primitives. Although it is not very hard to translate E (Zi), E (Zi) , E (Zi) and E (Zi) in ∞ Pν ¯ terms of the primitive ν. But E (ZiZi) is far from easy to translate. The paper is organized as follows. In section 2, we give some preliminaries about belief measures. In section 3, we give a unidirectional central limit theorem under belief measures and a central limit theorem for two-sided intervals is given in section 4.

2 Preliminaries

Let Ω be a Polish space and (Ω) be the Borel σ-algebra on Ω. We denote by (Ω) the compact subsets of B K Ω and ∆(Ω) the space of all probability measures on Ω. The set (Ω) will be endowed with the Hausdorff K topology generated by the topology of Ω.

Definition 2.1 A belief measure on (Ω, (Ω)) is defined as a set function ν : (Ω) [0, 1] satisfying: B B → (i) ν( )=0 and ν(Ω) = 1; ∅ (ii) ν(A) ν(B) for all Borel sets A B; ≤ ⊂ (iii) ν(Bn) ν(B) for all sequences of Borel sets Bn B; ↓ ↓ (iv) ν(G) = sup ν(K): K G, K is compact , for all open set G; { ⊂ }

3 (v) ν is totally monotone (or -monotone): for all Borel sets B , ..., Bn, ∞ 1 n I +1 ν( Bj ) ( 1)| | ν( Bj ) ≥ − i=1 =I 1,...,n j I [ ∅6 ⊂{X } \∈ where I is the number of elements in I. | |

Its conjugate ν∗ is given by c ν∗(A)=1 ν(A ), − where Ac is the complementary set of A. Note that, by Phillipe et al. [24], for all A (Ω), K (Ω) : K A is universally measurable. ∈ B { ∈ K ⊂ } Denote by u( (Ω)) the σ-algebra of all subsets of (Ω) which are universally measurable. The following B K K theorem belongs to Choquet [3]. We also refer to Phillipe et al. [24].

Theorem 2.2 The set function ν : (Ω) [0, 1] is a belief measure if and only if there exists a probability B → measure Pν on ( (Ω), ( (Ω))) such that K B K

ν(A)= Pν K (Ω) : K A , A (Ω). { ∈ K ⊂ } ∀ ∈B
Moreover, there exists a unique extension of Pν to ( (Ω), u( (Ω))). In the following, we still denote the K B K extension by Pν .

Let Ω∞ be the infinite product space of Ω, and (Ω∞) the Borel σ-algebra of Ω∞. For any A (Ω∞), B ∈B we define

ν∞(A)= P ∞ K = K K ... ( (Ω))∞ : K A , (2.1) ν { 1 × 2 × ∈ K ⊂ } where P ∞ ∆(( (Ω))∞) is the i.i.d. product probability measure. According
to Epstein and Seo [11] ν ∈ K e e (Lemma A.2.), ν∞ is the unique belief measure on (Ω∞, (Ω∞)) corresponding to P ∞. The conjugate of B ν ν∞ is denoted by ν∗∞.

Definition 2.3 The elements of a sequence ξi i∞=1 of random variables on (Ω, (Ω)) are called identically d { } B distributed w.r.t. ν, denoted by ξi = ξj , if for each i, j 1 and for all intervals G of R, ≥

ν(ξi G)= ν(ξj G). ∈ ∈ Remark 2.4 Note that belief functions are NOT additive in general, so they are NOT characterized by their values on intervals only. But we still use “all intervals” rather than “all Borel sets” in Definition 2.3 because our CLTs deal with intervals only.

In this paper, we are mainly interested in central limit theorems for random variables in the following form.

d Assumption 2.5 Let Yi ∞ be a sequence of bounded random variables on (Ω, (Ω)), Yi = Yj ,i,j 1. { }i=1 B ≥ We denote Xi(ω ,ω , ...) := Yi(ωi), i 1, (ω ,ω , ...) Ω∞. And let M be the least upper bound of Yi ∞ , 1 2 ≥ 1 2 ∈ { }i=1 that is, M = inf C 0 : ν( Yi C)=1 . { ≥ | |≤ }

4 Under Assumption 2.5, Xi ∞ is a sequence of random variables on (Ω∞, (Ω∞)) bounded by M. { }i=1 B It’s well known that the Choquet integral of Yi,i 1 w.r.t. belief measure ν, denoted by ν (Yi) is ≥ E M 0 ν (Yi)= ν(Yi t)dt + [ν(Yi t) 1]dt E 0 ≥ M ≥ − Z Z− M 0 = ν(Yi >t)dt + [ν(Yi >t) 1]dt,i 1. (2.2) 0 M − ≥ Z Z− In the remaining paper, we shall use equation (2.2) repeatedly without reference.

3 Unidirectional central limit theorem

This section mainly studies a central limit theorem for belief measure in a unidirectional interval. As we will show, the belief measures of the events where the partial sums of Xi ∞ is in unidirectional intervals can be { }i=1 approximated by a normal distribution in another probability space. But the belief measures of the events where the partial sums of Xi ∞ is in two-sided intervals are rather involved and can not approximated by { }i=1 a normal distribution. Actually it should be approximated by a bivariate normal distribution. So we leave the central limit theorem for a two-sided interval to the next section.

Lemma 3.1 Let Yi ∞ and Xi ∞ be defined in Assumption 2.5, then for any interval G of R, we have { }i=1 { }i=1

ν∞(Xi G)= ν(Yi G), i 1. ∈ ∈ ≥ Proof. In fact, by (2.1) and Theorem 2.2,

ν∞(Xi G)= P ∞ K = K K ... ( (Ω))∞ : K (ω ,ω , ...): Xi(ω ,ω , ...) G ∈ ν { 1 × 2 × ∈ K ⊂{ 1 2 1 2 ∈ }} = P ∞ K = K K ... ( (Ω))∞ : K (ω ,ω , ...): Yi(ωi) G
ν { e 1 × 2 × ∈ K e ⊂{ 1 2 ∈ }} = P ∞ K = K K ... ( (Ω))∞ : Ki ωi : Yi(ωi) G
ν { e 1 × 2 × ∈ K e ⊂{ ∈ }} = Pν Ki (Ω) : Ki ωi : Yi(ωi) G
{ e ∈ K ⊂{ ∈ }} = ν(Yi G), i 1. 
∈ ≥

Lemma 3.2 For Yi ∞ and Xi ∞ defined in Assumption 2.5, let { }i=1 { }i=1

Zi(K1 K2 ...) = inf Xi(ω1,ω2, ...), × × ωi Ki ∈ Z¯i(K1 K2 ...) = sup Xi(ω1,ω2, ...),Ki (Ω), i 1. × × ωi Ki ∈ K ≥ ∈

We claim that Z ∞ and Z¯i ∞ are two sequences of i.i.d. bounded random variables defined on { i}i=1 { }i=1 ( (Ω))∞, u(( (Ω))∞) w.r.t. P ∞. Moreover, K B K ν
P ∞ P ∞ E ν [Z ]= ν (Yi) =: µ, E ν [Z¯i]= ν∗ (Yi)=:µ, ¯ i 1, (3.1) i E E ≥ M ∞ 0 Pν 2 2 V ar [Zi]= 2tν(Yi t)dt + 2t[ν(Yi t) 1]dt µ =: σ , (3.2) 0 ≥ M ≥ − − Z Z− M ∞ 0 P 2 2 V ar ν [Z¯i]= 2tν∗(Yi t)dt + 2t[ν∗(Yi t) 1]dt µ =:σ ¯ , i 1. (3.3) 0 ≥ M ≥ − − ≥ Z Z−

5 Proof. We only prove the assertions for Z ∞ . The proofs for Z¯i ∞ are parallel. { i}i=1 { }i=1 By Assumption 2.5, M is the least upper bound of Xi ∞ in the sense that { }i=1

M = inf C 0 : ν∞( Xi C)=1 , i 1. { ≥ | |≤ } ≥

So Z ∞ is bounded by M. { i}i=1 For each i, j 1, t,ti,tj R, ≥ ∀ ∈

P ∞(Z t)= P ∞ K = K K ... ( (Ω))∞ : Z (K) t ν i ≥ ν { 1 × 2 × ∈ K i ≥ } = Pν∞ K = K1 K2 ... ( (Ω))∞ : inf Xi(ω1
,ω2, ...) t e ωi Kei { × × ∈ K ∈ ≥ }
= P ∞ Ke = K1 K2 ... ( (Ω))∞ : K (ω1,ω2, ...): Xi(ω1,ω2, ...) t ν { × × ∈ K ⊂{ ≥ }} = ν∞(Xi t)
e≥ e = ν(Yi t) ≥ = ν(Yj t) ≥ = ν∞(Xj t) ≥ = P ∞(Z t), ν j ≥ and for i = j, 6

P ∞(Z ti,Z tj ) ν i ≥ j ≥

= P ∞ K = K K ... ( (Ω))∞ : Z (K) ti,Z (K) tj ν { 1 × 2 × ∈ K i ≥ j ≥ }
= Pν∞ Ke = K1 K2 ... ( (Ω))∞ : infωei Ki Xi(ω1,ωe2, ...) ti, infωj Kj Xj(ω1,ω2, ...) tj { × × ∈ K ∈ ≥ ∈ ≥ }
= Pν∞ Ke = K1 K2 ... ( (Ω))∞ : infωi Ki Xi(ω1,ω2, ...) ti { × × ∈ K ∈ ≥ } ×
Pν∞ Ke = K1 K2 ... ( (Ω))∞ : infωj Kj Xj(ω1,ω2, ...) tj { × × ∈ K ∈ ≥ }
= P ∞(Z e ti)P ∞(Z tj ), ν i ≥ ν j ≥ where the third equality follows from the definition of product measure.

Similarly, we can easily get for any different positive integer ii, ..., in,

P ∞(Z ti1 , ..., Z ti )= P ∞(Z ti1 ) P ∞(Z ti ). ν i1 ≥ in ≥ n ν i1 ≥ ×···× ν in ≥ n

Thus, Z ∞ is a sequence of independent and identically distributed random variables under P ∞. { i}i=1 ν Moreover, we calculate the expectation and variation of Zi as follows: M ∞ 0 Pν E (Zi)= Pν∞(Zi t)dt + [Pν∞(Zi t) 1]dt 0 ≥ M ≥ − Z Z− M 0 = ν∞(Xi t)dt + [ν∞(Xi t) 1]dt 0 ≥ M ≥ − Z Z− M 0 = ν(Yi t)dt + [ν(Yi t) 1]dt 0 ≥ M ≥ − Z Z− = ν (Yi), i 1, E ≥

6 where the third equality is due to Lemma 3.1. And

∞ ∞ ∞ 2 Pν Pν 2 Pν V ar (Zi)= E (Zi ) E (Zi) 2 − M 2
2 = Pν∞(Zi t)dt µ 0 ≥ − Z 2 2 M M 2 = Pν∞(Zi √t)dt + Pν∞(Zi √t)dt µ 0 ≥ 0 ≤− − Z 2 Z 2 M M 2 = ν∞(Xi √t)dt + [1 ν∞(Xi √t)]dt µ ≥ − ≥− − Z0 Z0 M 0 2 = 2tν∞(Xi t)dt + 2t[ν∞(Xi t) 1]dt µ 0 ≥ M ≥ − − Z Z− M 0 2 = 2tν(Yi t)dt + 2t[ν(Yi t) 1]dt µ , i 1, 0 ≥ M ≥ − − ≥ Z Z− where the last equality is due to Lemma 3.1. 

2 2 Remark 3.3 In probability theory, E(Y ) (EYi) is a variant definition of variance. Under non-additive i − 2 2 probabilities, there are many ways to define the “variance” among which ν (Y ) ( ν (Yi)) seems intuitive. E i − E It’s easy to verify that for 0 Yi M, ≤ ≤ M 0 2 2 σ = 2tν(Yi t)dt + 2t[ν(Yi t) 1]dt µ 0 ≥ M ≥ − − Z Z− 2 2 = ν (Y ) ( ν (Yi)) . E i − E 2 2 2 Unfortunately σ = ν (Y ) ( ν (Yi)) for general Yi M. For example, readers can easily verify that for 6 E i − E | |≤ M Yi 0, − ≤ ≤ M 0 2 2 σ = 2tν(Yi t)dt + 2t[ν(Yi t) 1]dt µ 0 ≥ M ≥ − − Z Z− 2 2 2 2 = ν∗ (Y ) ( ν (Yi)) = ν (Y ) ( ν (Yi)) , E i − E 6 E i − E except that ν degenerates to a probability measure.

Let N( ) be the cumulate distribution function of a standard normal distribution. Our main result in · this section is the following central limit theorem.

Theorem 3.4 For Xi ∞ defined in Assumption 2.5, we have { }i=1 n Xi nµ i=1 − lim ν∞ α =1 N(α), (3.4) n P nσ →∞ √ ≥ ! − and n Xi nµ¯ i=1 − lim ν∞ < α = N(α), α R, (3.5) n P nσ →∞ √ ¯ ! ∀ ∈ where µ, µ,¯ σ, σ¯ are defined in Lemma 3.2.

7 Proof. By Theorem 2.2, we have

n ν∞ Xi α ≥  i=1  X n = P ∞ K = K K ... ( (Ω))∞ : K (ω ,ω , ...): Xi α ν { 1 × 2 × ∈ K ∈{ 1 2 ≥ }} i=1 X
e e n = P ∞ K = K K ... ( (Ω))∞ : K (ω ,ω , ...): Yi(ωi) α ν { 1 × 2 × ∈ K ∈{ 1 2 ≥ }} i=1 X
e e n = Pν∞ K = K1 K2 ... ( (Ω))∞ : inf Yi(ωi) α { × × ∈ K (ω1,ω2,...) K1 K2 ... ≥ } ∈ × × i=1 X
e n = Pν∞ K = K1 K2 ... ( (Ω))∞ : inf Yi(ωi) α { × × ∈ K ωi Ki ≥ } i=1 ∈ X
e n = P ∞ K = K K ... ( (Ω))∞ : Z (K K ...) α ν { 1 × 2 × ∈ K i 1 × 2 × ≥ } i=1 X
en = P ∞ Z α , α R. ν i ≥ ∀ ∈ i=1  X  Thus

n n Xi nµ Zi nµ i=1 − i=1 − R ν∞ α = Pν∞ α , α . (3.6) P √nσ ≥ ! P √nσ ≥ ! ∀ ∈

Applying the classical central limit theorem to Z ∞ deliver the corresponding limit theorem (3.4). { i}i=1 As for the proof of limit theorem (3.5), we just reverse the inequality sign and replace Z ∞ by Z¯i ∞ { i}i=1 { }i=1 in the preceding proof. 

4 Central limit theorem for two-sided intervals

In this section we will give the CLT for belief measures in two-sided intervals. We approximate the belief measures of the events where the partial sums of Xi ∞ is in a two-sided intervals by a bivariate normal { }i=1 distribution. Let N ( , ; ρ) be the cumulate distribution function for the bivariate normal distribution with zero means, 2 · · unit variances and correlation coefficient ρ, i.e.,

N (α , α ; ρ) = Pr(B α ,B α ), 2 1 2 1 ≤ 1 2 ≤ 2 where (B1,B2) is a bivariate normal random variable with the indicated moments.

Lemma 4.1 Under the assumption of Lemma 3.2, we have

2 M t2 ∞ M Mµ¯ + Mµ M M ν(t1 Yi t2)dt1dt2 µµ¯ Pν ¯ ρ := corr (Zi, Zi)= − − − − ≤ ≤ − . (4.1) R R σσ¯

8 Proof. Note that for positive random variables η and ζ,

∞ ∞ E[ηζ]= E I η x dx I ζ y dy 0 { ≥ } 0 { ≥ } h Z Z i ∞ ∞ = P (η x, ζ y)dxdy, ≥ ≥ Z0 Z0 where the second equality follows from Fubini’s theorem. Then

∞ ∞ ∞ ∞ ∞ P P + + P + P + P E ν [Z Z¯i]= E ν [Z Z¯ ] E ν [Z Z¯−] E ν [Z−Z¯ ]+ E ν [Z−Z¯−]= I + I + I + I , i i i − i i − i i i i 1 2 3 4 where

∞ Pν + ¯+ I1 = E [Zi Zi ] M M ¯ = Pν∞(Zi t1, Zi t2)dt1dt2 0 0 ≥ ≥ Z M Z M M M = P ∞(Z t )dt dt P ∞(Z t , Z¯i t )dt dt , ν i ≥ 1 1 2 − ν i ≥ 1 ≤ 2 1 2 Z0 Z0 Z0 Z0 P ∞ + I = E ν [Z Z¯−] 2 − i i M M = P ∞(Z t , Z¯i t )dt dt − ν i ≥ 1 − ≥ 2 1 2 Z0 Z0 0 M ¯ = Pν∞(Zi t1, Zi t2)dt1dt2, − M 0 ≥ ≤ Z− ∞ Z P + I = E ν [Z−Z¯ ] 3 − i i M M = P ∞( Z t , Z¯i t )dt dt − ν − i ≥ 1 ≥ 2 1 2 Z0 Z0 M 0 ¯ = Pν∞(Zi t1, Zi t2)dt1dt2 − 0 M ≤ ≥ Z Z− M 0 ¯ ¯ = Pν∞(Zi t2) Pν∞(Zi t1) Pν∞(Zi t1, Zi t2) dt1dt2, 0 M ≤ − ≤ − ≥ ≤ Z ∞Z− h i Pν ¯ I4 = E [Zi−Zi−] M M ¯ = Pν∞( Zi t1, Zi t2)dt1dt2 0 0 − ≥ − ≥ Z 0 Z 0 ¯ = Pν∞(Zi t1, Zi t2)dt1dt2 M M ≤ ≤ Z− Z− 0 0 ¯ ¯ = Pν∞(Zi t2) Pν∞(Zi t1, Zi t2) dt1dt2. M M ≤ − ≥ ≤ Z− Z− h i

9 Adding I1, I2, I3, I4 up, we get

∞ Pν ¯ E [ZiZi] M M M M ¯ = Pν∞(Zi t1, Zi t2)dt1dt2 + Pν∞(Zi t1)dt1dt2 − M M ≥ ≤ 0 0 ≥ Z− Z− Z Z M 0 0 0 ¯ ¯ + Pν∞(Zi t2) Pν∞(Zi t1)dt1dt2 + Pν∞(Zi t2)dt1dt2 0 M ≤ − ≤ M M ≤ Z Z− Z− Z− M M h M M ¯ ¯ 2 = Pν∞(t1 Zi Zi t2)dt1dt2 + M Pν∞(Zi t2)dt2 M Pν∞(Zi t1)dt1 + M − M M ≤ ≤ ≤ M ≤ − M ≤ Z− Z− Z− Z− M M M M ¯ 2 = ν∞(t1 Xi t2)dt1dt2 + M Pν∞(Zi t2)dt2 M Pν∞(Zi t1)dt1 + M − M M ≤ ≤ M ≤ − M ≤ Z− Z− Z− Z− M M 2 = ν(t1 Yi t2)dt1dt2 + M(M µ¯) M(M µ)+ M − M M ≤ ≤ − − − Z− Z− M t2 2 = M Mµ¯ + Mµ ν(t1 Yi t2)dt1dt2. − − M M ≤ ≤ Z− Z− It yields that

P ∞ P ∞ P ∞ ∞ E ν (Z Z¯i) E ν (Z )E ν (Z¯i) Pν ¯ i i ρ := corr (Zi, Zi)= − P ∞ 2 P ∞ 2 P ∞ 2 P ∞ 2 E ν (Z ) (E ν (Z )) E ν (Z ) (E ν (Zi)) i − i i − q2 M tq2 M Mµ¯ + Mµ M M ν(t1 Yi t2)dt1dt2 µµ¯ = − − − − ≤ ≤ − .  R R σσ¯

Theorem 4.2 For Xi ∞ defined in Assumption 2.5, there is a constant K which does not depend on { }i=1 α1, α2 or n, such that,

n K ν∞(α √nσ + nµ Xi α √nσ¯ + nµ¯) N2( α , α ; ρ) , 1 ≤ ≤ 2 − − 1 2 − ≤ √n i=1 X where ρ is defined in Lemma 4.1.

Moreover, the similar result holds if α1 and α2 depend on n.

Proof: Let Z ∞ and Z¯i ∞ be the same as in Lemma 3.2, and ρ be defined in Lemma 4.1. Then we { i}i=1 { }i=1 have

Zi µ Zi µ ∞ − ∞ − 1 ρ P σ P σ E ν   =0, Var ν   =   . Z¯i µ¯ Z¯i µ¯ − − ρ 1  σ¯   σ¯               

10 And we have, n ν∞(α √nσ + nµ Xi α √nσ¯ + nµ¯) 1 ≤ ≤ 2 i=1 X n = P ∞ K = K K ... ( (Ω))∞ : K (ω ,ω , ...): α √nσ + nµ Xi α √nσ¯ + nµ¯ ν { 1 × 2 × ∈ K ⊂{ 1 2 1 ≤ ≤ 2 }} i=1 X
e e n = P ∞ K = K K ... ( (Ω))∞ : K (ω ,ω , ...): α √nσ + nµ Yi(ωi) α √nσ¯ + nµ¯ ν { 1 × 2 × ∈ K ⊂{ 1 2 1 ≤ ≤ 2 }} i=1 X
e e n = Pν∞ K = K1 K2 ... ( (Ω))∞ : α1√nσ + nµ inf Yi(ωi) { × × ∈ K ≤ (ω1,ω2,...) K1 K2 ... ∈ × × i=1 X e n sup Yi(ωi) α2√nσ¯ + nµ¯ ≤ (ω1,ω2,...) K1 K2 ... ≤ } ∈ × × i=1 X
n n

= Pν∞ K = K1 K2 ... ( (Ω))∞ : α1√nσ + nµ inf Yi(ωi) sup Yi(ωi) α2√nσ¯ + nµ¯ { × × ∈ K ≤ ωi Ki ≤ ω K ≤ } i=1 ∈ i=1 i i X X ∈
e n n = P ∞ K = K K ... ( (Ω))∞ : α √nσ + nµ Z (K) Z¯i(K) α √nσ¯ + nµ¯ ν { 1 × 2 × ∈ K 1 ≤ i ≤ ≤ 2 } i=1 i=1 X X
e n n e e = P ∞(α √nσ + nµ Z (K) Z¯i(K) α √nσ¯ + nµ¯) ν 1 ≤ i ≤ ≤ 2 i=1 i=1 n X n X e¯ e Zi nµ Zi nµ¯ i=1 − i=1 − = Pν∞ α1 , α2 . (4.2) ≤ P √nσ P √nσ¯ ≤ !

′ Using similar method as in the proof of Lemma 3.2, we can easily get (Z , Z¯i) ∞ is a sequence of i.i.d. { i }i=1 random vectors under Pν∞. Thus, the classical central limit theorem goes through. 1 0 Let T =  . Then ρ 1  −√1 ρ2 √1 ρ2   − −    Z µ Z µ i− i− ∞ σ ∞ σ 1 0 EPν T   =0, VarPν T   =   , ¯ ¯ " Zi µ¯ # " Zi µ¯ # − − 0 1  σ¯   σ¯                and n n n P Zi nµ ¯ =1 − Zi nµ Zi nµ¯ i i=1 − i=1 − √nσ α1 , α2 T  n  C, (4.3) ≤ P √nσ P √nσ¯ ≤ ⇔ P Z¯i nµ¯ ∈  i=1 −   √nσ¯    for some convex C R2.   ⊂ According to the multidimensional Berry-Esseen Theorem [4], we get

n P Zi nµ i=1 − ˆ √nσ B1 K Pν∞ T  n  C Pr   C , (4.4) P Z¯i nµ¯ ∈ − ∈ ≤ √n =1 ! ˆ !  i −  B2  √nσ¯            11 Bˆ1 for some constant K, where   is the standard bivariate normal. Bˆ  2    Define  

B1 Bˆ1 1   = T −   . B Bˆ  2   2          By (4.3), we have

Bˆ1 Pr   C = Pr(α1 B1,B2 α2) = Pr( B1 α1,B2 α2)= N2( α1, α2; ρ). (4.5) ∈ ≤ ≤ − ≤− ≤ − − Bˆ !  2      Then Theorem 4.2 follows from (4.2), (4.3), (4.4), (4.5). This completes the proof. 

Remark 4.3 From equation (4.1), ρ depends on M. Actually, µ, µ,¯ σ, σ¯ depend on M too. Recall that

M is the least upper bound of Yi ∞ , that is, M = inf C 0 : ν( Yi C)=1 . So M is a parameter { }i=1 { ≥ | | ≤ } reflecting the distributional property of Yi ∞ . But we claim that if M is replaced by any number large than { }i=1 M in equation (4.1), the value of ρ does not change. For simplicity, we suppose Yi 0, i 1. In this case, ≥ ≥ M t2 P ∞ E ν (Z Z¯i)= Mµ ν(t Yi t )dt dt i − 1 ≤ ≤ 2 1 2 Z0 Z0 It’s enough to prove

M+1 t2 M t2 (M + 1)µ ν(t Yi t )dt dt = Mµ ν(t Yi t )dt dt . (4.6) − 1 ≤ ≤ 2 1 2 − 1 ≤ ≤ 2 1 2 Z0 Z0 Z0 Z0 And equation (4.6) follows from

M+1 t2 (M + 1)µ ν(t Yi t )dt dt − 1 ≤ ≤ 2 1 2 Z0 Z0 M t2 M+1 t2 = (M + 1)µ ν(t Yi t )dt dt ν(t Yi t )dt dt − 1 ≤ ≤ 2 1 2 − 1 ≤ ≤ 2 1 2 Z0 Z0 ZM Z0 M t2 M+1 M = (M + 1)µ ν(t Yi t )dt dt ν(t Yi)dt dt − 1 ≤ ≤ 2 1 2 − 1 ≤ 1 2 Z0 Z0 ZM Z0 M t2 = (M + 1)µ ν(t Yi t )dt dt µ − 1 ≤ ≤ 2 1 2 − Z0 Z0 M t2 = Mµ ν(t Yi t )dt dt . − 1 ≤ ≤ 2 1 2 Z0 Z0

Remark 4.4 If Xi(ω1,ω2, ...)= I ωi A1 , A1 (Ω), then { ∈ } ∈B

µ = ν(A ), µ¯ = ν∗(A ), σ = ν(A )(1 ν(A )), σ¯ = ν∗(A )(1 ν∗(A )), 1 1 1 − 1 1 − 1

12 2 P ∞ M Mµ¯ + Mµ ρ′ µµ¯ ρ = corr ν (Z , Z¯ )= − − − i i σσ¯ 1 t2 1 ν∗(A1)+ ν(A1) 1 1 ν(t1 Yi t2)dt1dt2 ν(A1)ν∗(A1) = − − − − ≤ ≤ − ν(A )(1 ν(A )ν (A )(1 ν (A )) 1 R −R 1 ∗ 1 − ∗ 1 1 0 1 ν∗(A1)+ νp(A1) 0 1 ν(Yi t2)dt1dt2 ν(A1)ν∗(A1) = − − − ≤ − ν(A1)(1 R νR(A1)ν∗(A1)(1 ν∗(A1)) − c − 1 ν∗(A )+ ν(A ) ν(A ) ν(A )ν∗(A ) = − 1 p 1 − 1 − 1 1 ν(A )(1 ν(A )ν (A )(1 ν (A )) 1 − 1 ∗ 1 − ∗ 1 ν(A )(1 ν∗(A )) = p 1 − 1 ν(A )(1 ν(A )ν (A )(1 ν (A )) 1 − 1 ∗ 1 − ∗ 1 ν(A )ν(Ac ) = p 1 1 . ν(A )(1 ν(A )ν (A )(1 ν (A )) 1 − 1 ∗ 1 − ∗ 1 By Theorem 4.2, p n

lim ν∞(α1√nσ + nµ Xi α2√nσ¯ + nµ¯) n ≤ ≤ →∞ i=1 X = N2( α , α ; ρ) − 1 2 −

B1 α1 = P    −  ≤ B α   2   2           ν(A1)(1 ν(A1))B1 ν(A1)(1 ν(A1))α1 = P  −   − −  , ≤ pν A ν A B pν A ν A α   ∗( 1)(1 ∗( 1)) 2   ∗( 1)(1 ∗( 1)) 2    −   −   p   p 

B1 1 ρ where   is the bivariate normal with zero mean and covariance matrix  −  . B ρ  2   1     −      ν(A1)(1 ν(A1))B1 Notice that  −  is a bivariate normal with zero mean and covariance matrix pν A ν A B  ∗( 1)(1 ∗( 1)) 2   −   p  ν(A1)(1 ν(A1)) ν(A1)ν(A2) Λ=  − −  which coincident with (1.4). ν A ν A ν A ν A  ( 1) ( 2) ( 2)(1 ( 2))   − −  Remark 4.5 When ν is additive, Theorem 4.2 degenerates into the classical central limit theorem, n Xi nµ i=1 − lim ν∞(α1 < α2) = Pr(α1

P ∞ E ν Xi = µ =µ, ¯ VarXi = σ =σ, ¯

13 and

M t2 ∞ Pν ¯ 2 E (ZiZi)= M Mµ¯ + Mµ ν∞(t1 Xi t2)dt1dt2 − − M M ≤ ≤ Z− Z− M t2 2 = M [ν∞(Xi t2) ν∞(Xi t1)]dt1dt2 − M M ≤ − ≤ Z− Z− M t2 M t2 2 = M ν∞(Xi t2)dt1dt2 + ν∞(Xi t1)dt1dt2 − M M ≤ M M ≤ Z− Z− Z− Z− M t2 M M 2 = M ν∞(Xi t2)dt1dt2 + ν∞(Xi t1)dt2dt1 − M M ≤ M t1 ≤ Z− Z− Z− Z M M 2 = M (t2 + M)ν∞(Xi t2)dt2 + (M t1)ν∞(Xi t1)dt1 − M ≤ M − ≤ Z− Z− M 2 = M 2 tν∞(Xi t)dt − M ≤ Z− M 2 2 = M ν∞(Xi t)dt − M ≤ Z− M M 2 2 2 = M t ν∞(Xi t) + t dν∞(Xi t) − ≤ M ≤ − M ∞ Z− = M 2 M 2 + EPν (X 2) − i P ∞ = E ν (X2), i 1. i ≥ ∞ ∞ ∞ ∞ 2 P ¯ P P ¯ Pν E ν (Z Zi) E ν (Z )E ν (Zi) E (Xi ) µµ¯ V arX i − i − i So ρ = ∞ 2 ∞ ∞ 2 ∞ = = = 1 and our theorem comes Pν Pν 2 Pν Pν 2 σσ¯ V arXi √E (Zi ) (E (Zi)) √E (Zi ) (E (Zi)) back to the classical− CLT. −

References

[1] M. Allais, Le comportement de l’homme rationel devant le risque: Critiqsuse des postulates et axiomes de l’ecole americaine, Econometrica, 21(1953), pp.503-546.

[2] Z. Chen, P. Wu and B. Li, A strong law of large numbers for non-additive probabilities, International Journal of Approximate Reasoning, 54(2013), pp.365-377.

[3] G. Choquet, Theory of capacities,Ann.Inst.Fouier(Grenoble), 5(1954), pp.131-295.

[4] A. Dasgupta, Asymptotic Theory of Statistics and Probability, Springer(2008).

[5] J. Dow and S, Werlang, Laws of large numbers for non-additive probabilities, Economics Working Pa- pers(Ensaios Economicos da EPGE), 226(1993).

[6] D. Ellsberg, Risk, ambiguity, and the Savage axioms, The quarterly journal of economics, 75(1961), pp. 643-669.

[7] L. Epstein and S.L.Ji, Ambiguous volatility and asset pricing in continuous time, The Review of Finan- cial Study, 26(2013), pp.1740-1786.

14 [8] L. Epstein and S.L.Ji, Ambiguous volatility, possibility and utility in continuous time, Journal of Math- ematical Economics, 50(2014), pp.269-282.

[9] L. Epstein, H. Kaido and K. Seo, Robust confidence regions for incomplete models, Econometrica, 84(2016), pp.1799-1838.

[10] L. Epstein and D. Schneider, IID: independently and indistingguishably distributed, Journal of Economic Theory, 113(2003), pp.32-50.

[11] L. Epstein and K. Seo, Exchangeable Capacities, Parameters and Incomplete Theories, Journal of Eco- nomic Theory, 157(2015), pp.879-917.

[12] F. Hu, Z. Chen, General laws of large numbers under sublinear expectations. Communications in Statistics-Theory and Methods, 45(2016), pp.4215-4229.

[13] P. Huber, V. Strassen, Minimax tests and the Neyman-Pearson lemma for capacities. The Annals of Statistics, 1(1973), pp.251-263.

[14] Z. Hu, L. Zhou, Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations. Acta Mathematica Sinica. English Series, 31(2015), pp. 305-318.

[15] M. Li, Y. Shi, A general central limit theorem under sublinear expectations. Science China Mathematics, 53(2010), pp.1989-1994.

[16] F. Macceroni and M. Marinacci, A strong law of large numbers for capacities. The Annals of Probability, 33(2005), pp.1171-1178.

[17] M, Marnacci, Limit laws for non-additive probabilities and their frequentist interpretation, Journal of Economic Theory, 84(1999), pp.145-195.

[18] S. Peng, G-expectation,G-Brownian motion and related stochastic calculus of Itoˆ type, Stochastic Anal- ysis and Applications, Abel Symposia, Springer,Berlin Heidelberg,(2007), pp.541-567.

[19] S. Peng, Muti-Dimensional G-Brownian motion and related Stochastic Calculus under G-Expectation, Stochastic Processes and their Applications, 118(2008), pp.2223-2253.

[20] S. Peng, A new central limit theorem under sublinear expectations. arXiv:0803.2656, (2008).

[21] S. Peng, Law of large numbers and central limit theorem under nonlinear expectations. arxiv:0702358, (2007).

[22] S. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochas- tic calculus under sublinear expectations. Science in China Series A-Mathematics, 52(2009), pp.1391- 1411.

[23] S. Peng, Nonlinear expectations and stochastic calculus under uncertainty with robust central limit the- orem and G-Brownian motion. arXiv:1002.4546, (2010).

15 [24] F. Philippe, G. Debs and J.Y. Jaffray, Decision making with monotone lower probabilities of infinite order. Mathematics of Operations Research, 24(1999), pp.767-784.

[25] D. Schmeidler, Subjective probability and expected utility without additivity. Econometrica, 57(1989), pp.571-587.

[26] P. Walley and T. Fine, Towards a frequentist theory of upper and lower probability. The Annals of Statistics, 10(1982), pp.741-761.

[27] Z. Wang and G. Klir, Generalized measure theory, Springer Science and Business Media(2010).

[28] D. Zhang, Z. Chen,A weighted central limit theorem under sublinear expectations. Communications in Statistics-Theory and Methods, 43(2014), pp.566-577.

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