次线性数学期望下的极限理论及其应用
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摘要
自从Choquet [13]提出容度概念后,人们对容度理论越来越感兴趣.因为在经济,统计学,工程学等领域有很多带有不确定性的问题,它们无法用传统的可加概率测度来准确预测或描述.现实中,概率可加性假设局限性明显,越来越多的人们开始舍弃可加概率这一传统工具,转而使用非可加(上,下)概率测度这一新兴工具来刻画带有不确定性的问题.事实上,早在1954年,Keynes[24]就发现了此类转变需求,从而创建了不确定概率理论.容度,这一非可加概率测度,就成为了刻画不确定性问题时一种十分合适的数学工具(参见Aug-ust in [1], Maccherroni和Marinacci [27], Doob [17], Schmeidler [33])鉴于金融数学和应用统计的广泛需求,非可加(上,下)概率/期望下随机变量的基本性质成为人们争相研究的学术热点.
众所周知,大数定律(LLN)在概率论与数理统计的发展和应用中发挥了至关重要的奠基作用,与此同时,我们发现有关非可加容度/期望下(强)大数定律的研究成果十分丰富,主要分为两个学术流派：一个是非可加概率流派,其特征是用非可加(不确定的)概率来刻画非可加概率下随机变量的频率属性.另一个是非线性期望流派,其特征是用非可加(上,下)期望来刻画非可加期望下随机变量的频率属性.尽管这两个流派在经典线性概率理论下是等价的,他们在非线性框架下是完全不同的,因为一个非线性期望通常是不能被相应的非线性概率唯一确定的(参见Chen.Z, Chen.T和Davison [7]以及Choquet, Hu, Memin和Peng [12])非可加概率流派成果众多,比如早期经典文献Dow和Verlang [18]以及Walley和Fine [36],近期著名成果例如Cooman和Miranda [15], Epstein和Schneider [19], Marinacci [28], Mac-cheroni和Marinacci [27], Chen和Wu [10], Chen, Wu和Li [11]以及Terdn [34].在对非可加概率的不同假设下,文献证明了当实验次数增加时,通过大量实验得到的实验均值不再逼近于某个确定的期望值,而是在下概率(容度)下落在某个期望值区间内.
在非可加期望流派,Peng是第一个提出g-期望和G-期望的学者.在9-期望的启迪下,Peng[29,30,31]提出了次线性期望下随机变量的独立和同分布定义,我们称之为Peng独立.在某些对非可加期望的假设下,Peng通过偏微分方程(PDE)理论给出了次线性期望下的大数定律和中心极限定理.
将非可加概率和非可加期望两个学派的成果与经典大数定律相比较,我们发现,在对概率和期望的公理化性质进行削弱的同时,我们必须对状态空间,非可加概率,随机变量做出额外的技术性假设作为补偿.
自然地,我们想到如下问题：可否借鉴经典可加概率(Feller,Linderberg等人)的证明方法,从概率的角度将大数定律推广到次线性期望下呢?答案是肯定的.本文中,我们首先借鉴了经典的Linderberg-Feller的证明思想在事件独立下给出了Choquet期望下的大数定理,然后推广到更一般情形：卷积独立下的次线性期望下大数定律,进而获得容度下的弱大数定律,并给出了一个与二者相关的等价定理.本文证明过程仅使用了泰勒展开式,次线性期望性质等纯概率基础工具,未采用特征函数或PDEs等复杂辅助手段.进一步,与文献相比,我们削弱了大数定理的假设条件.比如,在次线性期望下,降低了随机变量的阶矩条件,随机变量独立性假设也弱化为卷积独立.此外,我们的次线性大数定律能够涵盖著名的Ellsberg模型(带有模糊性的罐子模型),而Ellsberg模型因为具有概率模糊性并不符合文献中大数定律的假设条件.
本文共分为四章.第一章研究在事件独立定义下,Choquet期望下的大数定律.第二章研究在卷积独立定义下,一般次线性期望下的大数定律.第三章给出次线性大数定律在模糊条件下的应用.第四章给出了次线性大数定律的收敛误差估计.
引入符号：
假设Ω为状态空间,F为σ-域.称函数X:Ω→R为可测空间(Ω,F)上随机变量,若X是F-可测的.令H为可测空间(Ω,F)上随机变量全体的子集.
假设Gb(R)为R上所有有界连续函数的集合C+b(R)为Cb(R)上的非负单调函数的全体C2b(R)为Cb(R)上那些函数一阶,二阶导数都存在且导数仍在Gb(R)中的函数全体.对给定有限常数μ和μ,记集合Dn:={y|y=(y1,y2,…,yn),yi∈[μ,μ],1≤i≤n}.
(Ⅰ)第一章主要研究上Choquet期望下的大数定律.
容度和Choquet期望在定义形式,基本性质等方面与概率和线性期望具有高度相似性.可以说,Choquet理论是联系经典线性理论与新兴次线性理论的桥梁.1999年到2005年,Maccherroni和Marinacci [27][28]提出了容度下的随机变量独立性定义,独立形式与概率下的独立定义相似.正是考虑到Choquet期望与线性期望的这种相似性,在将大数定律向非可加期望领域推广时,我们优先选择Choquet期望作为尝试的起点.所以本文先讨论Choquet这种相对简单的情况(此时类似线性期望有较多借鉴之处),在事件独立的假设下,给出了Choquet期望下大数定律.然后再考虑一般的次线性期望下大数定律.从简入难.
回顾Choquet期望下大数定律的理论进展,我们发现在事件独立的假设下,不同文献给出了不同技术假设,但从证明方式来看,主要分为以下两种：一种是转化法：“曲线救国”,比如Chareka [3]将不可加的Choquet积分转化成可加的Lebesgue-Stieltjes积分.然后利用Lebesgue-Stieltjes积分性质证明了Choquet框架下的强(弱)大数定律.另一种是直接证明法：比如Li和Chen[26]直接证得容度下的Chebyshev不等式和Borel-Cantelli引理,从而证明了Choquet期望下大数定律,证明方法类似线性LLN.由此启发我们：在事件独立下,可否将大数定律的其他(Linderberg, Feller等)经典证法推广到Choquet期望下证得大数定律呢?我们的答案是肯定.
我们知道,证明大数定律的关键条件是概率/期望的可加性和随机变量的阶矩条件.而本章讨论的Choquet期望恰为非可加期望.为了解决这个期望非可加问题,我们采用2-alternating容度,因为由2-alternating容度生成的Choquet期望具有我们所需的次可加性.在这个前提假设下,本章讨论了独立同分布随机变量序歹(?){Xi}∞i=1的依分布收敛(分布极限)问题.此外,对比其他Choquet结论,我们的Choquet大数定律对随机变量的阶矩条件也进行了弱化.
在引入大数定律前,我们先叙述三个核心引理作为铺垫.和变通项引理：
引理1.3.1令V为F上的2-alternating容度,且Cv,Cv分别为由其生成的上,下Choquet期望.令{Xi}i=1∞为(Ω,F)上的一列独立随机变量.则对任意单调函数φ∈Gb(R)和任意常数yi∈R,其中n
泰勒展开引理：引理1.3.2令V为2-alternating容度,Cv,Cv分别为其生成的上,下Choquet期望令{Xi)i=1∞为同分布随机变量列且满足Cv[Xi]=μ和Cv[Xi]=μ假设对任意i≥1,有Cv[|Xi|]<∞.则对任意函数φ∈Cb2(R),存在一个正值常数b。(∈)使得bn((?))→0当n→∞时,从而(Ⅰ)∑i=1n supx∈R{Cv[φ(x+xi/n)]-φ(x)≤supx∈R G(φ'(x),μ,μ)+bn((?)).(Ⅱ)∑i=1n infx∈R{Cv[φ(x+Xi/n)]-φ(x)≥infx∈R G(φ'(x),μ,μ)-bn((?)).其中G(x,μ,μ):=x+μ-x-μ.
引理1.3.3令G(x,y,z)函数定义同引理1.3.2,即G(x,y,z):=x+y-x-z.则对单调的φ∈Cb(R),有(Ⅰ)infy∈Dn supx∈R G(φ'(x),μ-1/n∑ni=1yi,μ-1/n∑i=1n yi)=0.(Ⅱ)infy∈Dn infx∈R G(φ'(x),μ-1/n∑ni=1yi,μ-1/n∑i=1n yi)=0.
经过上述三个引理的铺垫,我们可以引入本章第一个定理：分布极限定理.此定理表明,在Choquet期望下实验均值的分布极限是一个最大分布.
定理1.4.1(分布极限定理)令V为F上2-alternating容度,Cv,Cv分别为由其生成的上,下Choquet期望.假设{Xi}∞i=1为独立同分布随机变量列且满足Cv[Xi]=μ,Cv[Xi]=μ记部分和Sn:=∑ni=1Xi假设对任意i≥1,Cv[Xi]<∞.则对任意单调函数φ∈Cb(R),
定理1.4.2(容度下的弱大数定律)令V为F上2-alternating容度,Cv,Cv分别为由其生成的上,下Choquet期望.令v(A):=Cv[IA],(?)4∈F假设{Xi}∞i=1为独立同分布随机变量列且满足Cv[Xi]=μ,Cv[Xi]=μ.记Sn:=∑ni=1Xi假设对任意i>1,Cv[|Xi|]<∞若对任意φ∈Cb+(R),任意∈>0,则有
下面,我们给出一个令最大分布等价于容度下弱大数定律的充分条件.
定理1.4.3(等价定理)令V为F上2-alternating容度,Cv,Cv分别为由其生成的上,下Choquet期望.给定函数φ∈Cb+(R),假设{Xi}∞不=1为一列独立同分布随机变量列并满足Cv[Xi]=μ,Cv[Xi]=μ假设对任意i>1,Cv[|Xi|]<∞.令Sn:=∑ni=1Xi则结论(A)与(B)等价.(A)对任意(?)>0,令v(4):=Cv[IA],VA∈F,有(?)v(μ-ε≤Sn/n≤μ+ε)=1.(B)对任意φ∈Cb(R),
等价定理的意义在于：若收敛结论对单调φ∈Cb(R)成立,则对任意φ∈Gb(R)都成立.我们由分布极限定理推广得到如下Choquet期望下大数定律.
定理1.4.4(Choquet期望下大数定律)令V为F上2-alternating容度,Cv,Cv分别为由其生成的上,下Choquet期望.假设{Xi}∞i=1为独立同分布随机变量列且满足Cv[Xi]=μ,Cv[Xi]=μ假设对任意i≥1,Cv[|Xi|]<∞.记Sn:=∑ni=1,Xi则对任意函数φ∈Cb(R),有
注1.4.5由定理证明过程可知,随机变量同分布条件可以削弱为:随机变量具有有限共一阶矩,即{Xi}∞i=1满足1≤i≤n, Cv[Xi]=Cv[Xi], Cv[Xi]=Cy[X1]； Cv[|XI|]=Cv[|x1|], Cv[|Xi|]=Cv[|X1|]<∞.
(Ⅱ)在第二章,我们用概率语言证得了次线性期望下的大数定律.
本章是经典大数定律的自然推广.本章有四个主要结论：(1)我们对广义Ells-berg模型的极限分布进行了研究,并发现它的极限分布是一个最大分布.(2)我们将Ellsberg模型推广,得到了一个有关随机变量的充分条件,在这个条件下,实验均值的极限分布与Ellsberg模型的极限分布是一致的.(3)在次线性期望的φ-卷积独立定义下,我们给出了最大分布等价于容度下弱大数定律的充分条件.(4)我们将本章结论与文献结论进行了对比.在线性期望下卷积独立的启发下,我们将卷积独立这个概念推广到次线性期望之下.和变通项引理：
引理2.3.1给定函数φ∈Cb(R)假设E为次线性期望,ε为其共轭期望.令{Xi}i=1∞为E下一列φ-卷积独立随机变量.则对任意常数yi∈R,1≤i≤n,有其中n则上述引理有如下变形：引理2.3.2令P为概率测度集,{Xi)i=1∞在每一个概率Q∈P下都是一列独立随机变量.则对任意常数yi∈R,i=1,2,…,n,和任意函数φ∈Cb(R),有其中
下述泰勒展开引理是证明Choquet大数定律时的核心引理.我们将(0.1)式视为次线性期望E下的Linderberg条件.
引理2.3.3假设E为次线性期望,￡为其共轭期望.假设随机变量序列{Xi}i=1∞具有有限共一阶矩且E[Xi]=μ,ε[Xi]=μ假设对任意∈>0,有则对任意单调函数φ∈C2b(R),(Ⅲ)特别的,若E[·]和ε[·]为概率测度集P上的上,下期望算子且满足
则对任意单调函数φ∈C2b(R),经过上述引理的铺垫,我们引出容度下/次线性期望下的大数定律.
定理2.4.1(Ellsberg型大数定律)给定一个概率测度集P,令(E,ε)分别为P上EQ生成的上,下期望.假设对任意Q∈P,{Xi}∞i=1是Q下一列独立随机变量,{Xi}i=1n具有有限共一阶矩(?)μ:=E[Xi],μ:=ε[Xi]使得假设条件(0.1)成立.记Sn:=∑ni=1Xi进一步,若则(Ⅰ)对任意单调函数φ∈Cb(R),我们有F,则对任意∈>0,
定理2.4.2(次线性期望下的大数定律)假设E是一个次线性期望,￡是一个共轭期望.假设随机变量序列{Xi}∞i=1具有有限共一阶矩且E[Xi]=μ和ε[Xi]=μ.假设条件(0.1)成立.记部分和Sn:=∑ni=1Xi则有
(Ⅰ)给定单调函数φ∈Cb(R),若{Xi}∞i=1是E下一列φ-卷积独立随机变量,则
(Ⅱ)若对任意φ∈C+b(R),{X}∞i=1是E下一列φ-卷积独立随机变量,令v(A)：=ε[IA],VA∈(?),则对任意∈>0,
类似第一章Choquet期望的结构,Ellsberg型大数定律(定理2.4.1)和次线性期望下大数定律(定理2.4.2)中的函数φ都局限于Cb(R)上的单调函数,为此我们引入次线性期望下的等价定理(定理2.4.3),将对单调φ∈Cb(R)成立的定理推广到对任意φ∈Cb(R)成立.
定理2.4.3(次线性期望下的等价定理)假设E是一个次线性期望,ε是它共轭期望.对函数φ∈C+b(R),假设{Xi}∞i=1是一列φ-卷积独立的随机变量,具有有限共一阶矩且E[Xi]=μ和ε[Xi]=μ使得假设条件(0.1)成立.令Sn：=∑ni=1Xi则结论(A2)与(B2)等价.
似(A2)对任意∈>0,(B2)对任意φ∈Cb(R),
通过等价定理可知,Ellsberg型/E下的大数定律(Ⅰ)对任意φ∈Cb(R)都成立.表述如下.
定理2.4.4假设条件同定理2.4..1.则对任意函数φ∈Cb(R),有
定理2.4.5假设条件同定理2.4.2.给定任意φ∈Cb(R),若{Xi}∞i=1.是E下一列φ-卷积独立随机变量,则
注2.4.6若{Xi)∞i=1的阶矩阶数大于1,即对任意常数β>1,E[|Xi|β<∞.注意到如果supi≥1E[|Xi|β]<∞,则定理2.4.5和引理2.3.3中的假设条件(0.1)都成立.所以根据Peng [32]中引理3.9的证明,定理2.4.5的条件φ∈Cb(R)就可以弱化为:连续函数φ满足增长条件|φ(x)|≤C(1+|x|β-1),去掉有界性.
(Ⅲ)第三章,次线性大数定律在模糊条件下的应用.
例子3.1(带有模糊性的罐子模型)考虑有限可数个罐子,按顺序将编号记为{1,2,…}.实验者被告知第i个罐子里有100i个小球(此后100i表示100乘以i),颜色为红色或者黑色.第i个罐子里红色球的个数为25i到50i个不等.实验者不知道除此之外的任何信息.每一次只能从一个罐子里取出一个小球.记由定理2.4.4知:当实验次数足够多,n次试验中摸到红球的个数的实验均值服从如下最大分布
例子3.2(期权定价模型)令{B)t≥0为概率空间(Ω,F,P)上的几何布朗运动.{St≥o是服从几何布朗运动的股票价格:dSt=μStdt+σStdBt.在非完全市场下,欧式期权的未来损益函数为φ(ST):=(ST-L)+因为上期望是eμ+σκ,下期望是eμ-σκ,由定理2.4.5知股价的分布极限如下
(Ⅳ)第四章,主要研究次线性大数定律的收敛误差估计.
定理4.1假设E是一个次线性期望,￡是一个共轭期望.假设随机变量序列{Xi)∞i=1具有有限共一阶矩,E[Xi]=-μ,ε[Xi]=μ.若记部分和Sn:=∑ni=1Xi则二阶矩下的大数定律收敛误差估计如下.其中μ:=|μ|∨|μ|.
当阶矩条件降低至(?)supl≤i≤n E[|Xi|1+α]<∞,0<α<1,误差估计相关结果见定理4.2.
Ever since the definition of capacity was introduced by Choquet [13], it has been a heated scientific subject worldwide. Since in many applica-tion fields, such as finance, economics and robust statistics, the traditional additive probability measures fail to provide adequate or good information to describe or interpret the uncertain phenomena accurately. Therefore, in some areas, the assumption that given a precise probability for random vari-ables as an apparently quite natural property has been abandoned, in favor of non-additive/imprecise probabilities (such as lower-upper probabilities) or nonlinear expectations (for example Choquet expectations, sublinear expec-tations, lower-upper expectations). Indeed, as early as1954, Keynes [24] had discovered this problem, then constructed a theory of imprecise prob-ability. Meanwhile, capacities, the non-additive probability measures, seem to be a powerful tool to model the uncertainty when the assumption of addi-tivity is suspect.(e.g., Augustin [1], Maccherroni and Marinacci [27], Doob [17], Schmeidler [33]). Later, motivated by mathematical finance and ro-bust statistics, the frequentist properties of random variables for non-additive (lower and upper) probabilities have attracted more and more attentions.
As we known, the law of large numbers (LLN) plays an important role in probability theory and mathematical statistics. Meanwhile, in the literature, various generalizations of (strong) LLNs for non-additive probabilities have been established. Among them turn into two groups:one is called nonad-ditive probability group in which people like to use nonadditive (imprecise) probability to accommodate the frequentist properties of random variables for non-additive probability, the other is called nonlinear expectations group in which people like to use nonadditive (lower and upper) expectations to accommodate the frequentist properties of random variables for non-additive expectations. Although both groups in the framework of linear probability theory are equivalent, they are totally different in the nonlinear case in the sense that a nonlinear expectation usually could not be determined uniquely by the corresponding nonlinear probability (see for example,[7],[12]). In nonadditive probability group, the earlier papers by Dow and Werlang [18] and Walley and Fine [36], while the more recent results by Cooman and Miranda [15], Epstein and Schneider [19], Marinacci [28], Maccheroni and Marinacci [27], Chen and Wu [10], Chen, Wu and Li [11] and Teran [34]. Under various assumptions on nonadditive probabilities, they proved that the frequentist obtained from a large number of trials is no longer close to an expected value but an interval of possible expected value under lower probability as more trials are performed.
In nonadditive expectation group, Peng is the first to introduce the notions g-expectation and G-expectation. Inspired by g-expectation, Peng [29,30,31] introduces a new notion of identical and independently distributed (ⅡD) random variables under sub-linear expectations, which is called Peng independence. Under some assumptions on nonadditive expectations, he shows LLNs by using partial differential equations (PDEs).
Comparing the results obtained by the two groups with the classical LLN, one could find that the weakening of axiomatic properties of probability and expectation has been balanced by the incorporation of the extra technical assumptions on the state space and/or nonadditive probabilities and/or random variables.
A natural question is arising:Can we extend the LLN to the sub-linear expectations case by traditional pure probabilistic (Linderberg-Feller's) method, without using the characteristic functions or PDEs? The answer is affirmative. In this paper, we first extend the classical LLN to the Choquet expectation case with event independence under capacity. Then we extend LLN to the sub-linear expectation case with convolutionary random vari- ables. In both cases, we establish the equivalence theorem to connect the "distribution theorem" with the "Weak LLN under capacity". The whole proving method we proposed here only depends on some elementary prob-abilistic techniques, such as Taylor expansions and the basic properties of sub-linear expectations, without using any artificial tools like characteristic functions or PDEs. In this sense, our LLN is a natural extension of the classical LLN. Further, compared to other literature, our non-additive ver-sion of LLN has weaken the assumptions of the theorems, for example, we weaken the moment condition of the random variables, as well as replace the independence condition by the weaker notion of φ-convolution on ran-dom variables. Meanwhile, Ellsberg model satisfies our theorem assumptions.
This paper is divided into4chapters. In chapter1, under the event inde-pendence for capacity, we establish law of large numbers for Choquet expec-tations induced by2-alternating capacities. In Chapter2, we first introduce the notion of convolutionary random variables under sub-linear expectations, then prove the LLNs under both sub-linear expectations and capacities. In Chapter3, we give three application examples of our LLNs, especially the Ellsberg model(Urn model with ambiguity). The corresponding estimation of the convergence error of our LLNs is given in Chapter4.
Notations: Suppose that Ω is a state space and (?) is a σ-field. Function X:Ω→R is called a random variable on measurable space (Ω,(?)), if X is F-measurable. Let H be a subset of all random variables on measurable space (Ω,F).
Suppose that Cb(R) is the set of all bounded and continuous real-valued functions on R and Ckb(R) is the set of bounded and k-time continuously differentiable functions with bounded derivatives of all orders less than or equal to k. C+b(R) is all the non-negative monotonic functions in Cb(R)
For given finite constants μ and μ, set Dn:={y:=(y1,y2,…,yn):yi∈[μ,μ], i=1,2,…,n}.
(I) In Chapter1, we study the law of large numbers under Choquet expectations.
As we known, the definitions and properties of capacity/Choquet ex-pectations are quite similar to those of probability/linear expectations, thus, the Choquet theory can be viewed as a bridge well connecting the traditional probability theory with the new-arising capacity theory. From1999to2005, Maccherroni and Marinacci [27][28] introduce event independence under ca-pacity, which is in accord with the probability case. Due to this similarity, we extend the classical LLN starting from the Choquet expectation case, then consider the more general and complex case:the sub-linear expectation case.
Reviewing the literature of LLN under Choquet expectations, different authors propose different assumptions of LLN. However, the main pattern of proving LLN can be divided into2modes:one is the indirect method, such as Chareka [3], he turned the non-additive Choquet integral into the additive Lebesgue-Stieltjes integral, then by using the existing properties of Lebesgue-Stieltjes integral he derived the LLN under Choquet expectations. The other is the direct method, such as Li and Chen [26], they obtained LLN by proving Chebyshev inequality and Borel-Cantelli lemma under capacity, still following the proving pattern of traditional LLN. These inspire us with a rising question:With event independence under capacity, can we extend the LLN to Choquet expectations by pure probabilistic (Linderberg-Feller's) way? The answer is affirmative.
Recall that the key to prove classical LLN are the additivity of proba-bilities/expectations as well as the moment conditions of random variables. However, the Choquet expectations happen to be the non-additive ones. To overcome this non-additive problem, we adopt the2-alternating capacity, thus the Choquet expectations induced by it turn to be sub-additive ones, under which we consider the sequence{Xi}∞i=1of independent and identically distributed (ⅡD) random variables converges in distribution. Moreover, our moment condition on random variables is weaker than other literature.
Lemma1.3.1Let V be a2-alternating capacity defined on (?),and Cv,Cv be the induced upper,lower Choquet expectation respectively.Let {Xi}i=1∞be a sequence of independent random variables.Then for any monotonic φ∈Cb(R) and any constant yi∈R,
Lemma1.3.2Let V be a2-alternating capacity and Cv,Cv be the induced upper,lower Choquet expectation respectively.Let {Xi}i=1∞be a sequence of identical distributed random variables with Cv[Xi]=μ and Cv[Xi]=μ satisfying that for i≥1, Cv[|Xi|]<∞.
Then,for each function φ∈Cb2(R),there exists a positive constant bn(∈)with bn(∈)→0,as n→∞,such that (Ⅰ)∑i=1n supx∈R{Cv[φ(x+Xi/n)]-φ(x))≤supx∈R G(φ'(x),μ,μ)+bn(∈).(Ⅱ)∑i=1n,infx∈R{Cv[φ(x+Xi/n)]-φ(x)}≥infx∈R G(φ'(x),μ,μ)-bn(∈). Where G(x,μ,μ):=x+μ-x-μ.
Lemma1.3.3Let G(x,y,z) be the function defined in Lemma1.3.2,that is G(x,y,z):=x+y-x-z. Then for any monotonic φ∈Cb(R),
(Ⅰ)infy∈Dn supx∈R G(φ'(x),μ-1/n∑i=1n yi,μ-1/n∑i=1n yi)=0. (Ⅱ)infy∈Dn infx∈R G(φ'(x),μ-1/n∑ni=1yi,μ-1/n∑ni=1yi)=0.
Our new LLNs under Choquet expectations are stated as follows. Theorem1.4.1(Limit Distribution Theorem)Let V be a2-alternating capacity defined on (?),and Cv,Cv be the induced upper, lower Choquet expectation respectively.Let {Xi}∞i=1be a sequence of ⅡD random variables on (Ω,(?))with Cv[Xi]=μ,Cv[Xi]=μ.ASSume that for i≥1, Cv[|Xi|]<∞Set Sn:=∑ni=1Xi. Then for each monotonic function φ∈Cb(R),(Ⅰ)(?)Cv[φ(Sn/n)]=supμ≤x≤μ φ(x)；(Ⅱ)(?)Cv[φ(Sn/n)]=infμ≤x≤μ φ(x).
The limit distribution theorem indicates that the limit distribution of the empirical average of random variables is the maximal distribution.
Theorem1.4.2(Weak LLN under capacity)Let V be a2-alternating capacity defined on (?),and Cv,Cv be the induced upper,lower Choquet ex-pectation respectively.Let v(A):=Cv[IA],(?)A∈(?).Let{Xi}∞i=1be a sequence of ⅡD random variables with Cv[Xi]=μ,Cv[Xi]=μ.Assume that for i≥1, Cv[|Xi|]<∞. Set Sn:=∑ni=1Xi. If for any function φ∈C+b(R),any (?)>0,then
Theorem1.4.3(Equivalence Theorem)Let V be a2-alternating capacity defined on (?),and Cv,Cv be the induced upper,lower Choquet expectation respectively.Given function φ∈C+b(R),Let{Xi)∞i=1be a sequence of ⅡD random variables with Cv[Xi]=μ,Cv[Xi]=μ.Assume that for i≥1, Cv[|Xi|]<∞. Let Sn:=∑ni=1Xi.Then the followings are equivalent. (A) For any (?)>0, let v(A):=Cv[IA],(?)A∈F, then (B) For any φ∈Cb(R),
The equivalence theorem states that, if the convergent result is true for any monotonic function φ∈Cb(R), then it is still true for any φ∈Cb(R)
Theorem1.4.4(LLN under Choquet expectations) Let V be a2-alternating capacity defined on (?), and Cv, Cv be the induced upper, lower Choquet expectation respectively. Let{Xi}∞i=1be a sequence of IID random variables with Cv[Xi]=μ, Cv[Xi]=μ. Assume that for i≥1, CV[|Xi|]<∞. Set Sn:=∑ni=1Xi.Then for each function φ∈Cb(R),
Remark1.4.5Further, the condition of identical distribution in above the-orems can be weaken to "finite common first moment condition", that is, for1≤i≤n, Cv[Xi]=Cv[X1], Cv[Xi]=Cv[X1]; Cv[|Xi|]=Cv[|X1|], Cv[|Xi|]=Cv[|X1|]<∞.
(Ⅱ) In Chapter2, we prove the LLNs under sub-linear ex-pectations by using pure probabilistic method, without using the characteristic functions or PDEs. It is a nature extension of the tradition LLNs.
We obtain four principal results in this chapter.(1) We explore the limit distribution of general Ellsberg-type model mentioned above and show that its limit distribution is a maximal distribution.(2) We extend Ellsberg-type model to more general case, and obtain a sufficient condition on ran-dom variables and sub-linear expectations, under which the empirical average of random variables has the same limit distribution as Ellsberg-type model does.(3) With a new notion of φ-convolution on sub-linear expectations and random variables, we show both maximal distribution and weak LLNs are equivalent.(4) We compare our results with which appearing in the men-tioned articles.
Lemma2.3.1Let E be a sub-linear expectation and ε be its conjugate ex-pectation. Given a function φ∈Cb(R). Let{Xi}∞i=1be a sequence of φ-convolutionary random variables under E. Then for any constant yi∈R, i=1,2,…,n,
If sub-linear E is the upper expectation operator generated by a set P of probability measures, that is E[·]:=(?)Eq[·], then Lemma2.3.1can be restated as follows.
Lemma2.3.2Let P be a set of probability measures and{Xi}∞I=1be a sequence of independent random variables for each Q∈P. EQ is the linear expectation with respect to probability Q. Then for any constant yi∈R., i=1,2,…,n,and any function φ∈Cb(R), where
The following lemma is a key to the proof of our LLN.Condition(0.2) can be viewed as the Linderberg condition under sub-linear expectation E.
Lemma2.3.3Let E be a sub-linear expectation and ε be its conjugate ex-pectation.Suppose that{Xi}i=1∞is a sequence of random variables with finite common first moment and E[Xi]=μ,ε[Xi]=μ,i≥1.Moreover,for any (?)>0,if then for any monotonic function φ∈Cb2(R),we have (Ⅰ)limn→∞infy∈Dn∑i=1n supx∈R{E[φ(x+Xi-yi/n)]-φ(x)}=0；(Ⅱ)limn-→∞infy∈Dn∑i=1n infx∈R{E[φ(x+Xi-yi/n)]-φ(x))=0.(Ⅲ)Moreover,if E[·] and ε[·] are the upper and lower expectations on the
set P of probability measures such that
then,for any monotonic function φ∈Cb2(R),
We are now ready to prove the LLN for sub-linear expectations and ca-pacities. Similar to the structure of Chapter1, We first present the Ellsberg-LLN(Theorem2.4.1),then the sublinear LLN(Theorem2.4.2).
Theorem2.4.1(Ellsberg-LLN) Given a set P of probability measures, let E, ε be the upper and lower expectations of EQ over P respectively. Assume that for any Q∈P,{Xi}∞i=1is a sequence of independent random variables under Q, and{Xi}ni=1has the finite common first moment μ:=E[Xi] and μ:=ε[Xi] such that condition (0.2) holds. Set Sn:=∑ni=1Xi.Furthermore, if (Ⅰ) For any monotonic φ∈Cb(R), then (Ⅱ)For φ∈C+b(R). Set V(A):=(?)Q(A), v(A):=(?)Q(A),(?)A∈F, then for any (?)>0,
Next, we extend the Ellsberg-LLN with event independence to the LLN under sub-linear expectation E with φ-convolutionary random variables. It is worthwhile to note that the limit distribution of the empirical average under E is still a maximal distribution.
Theorem2.4.2(LLN for sub-linear expectations) Let E be a sub-linear expectation and ε be its conjugate expectation. Assume that{Xi}∞i=1is a sequence of random variables with finite common first moment-E[Xi]=μ
(Ⅰ)Given monotonic φ∈Cb(R),if {Xi}∞i=1is a sequence of φ-convolutionary random varables under E,then
(Ⅱ)If for any φ∈Cb+b(R),{Xi}∞i=1is a sequence of φ-convolutionary randorn variables under E,then for any (?)>0,and for v(A):=ε[IA],(?)A∈F, we have
Since the function φ in both LLNs is limited to the monotonic φ∈Cb(R),we need the following equivalence theorem to extend the monotonic φ∈Cb(R) to all φ∈Cb(R).
Theorem2.4.3(Equivalence Theorem under E)Let E be a sub-linear expectation and εbe its conjugate expectation.Given function φ∈Cb+(R), assume that {Xi}∞i=1is a sequence of φ-convolutionary random variables with finite common first moment E[Xi]=μ and ε[Xi]=μ.If condition (0,2)(A2) For any (?)>0,v(A):=ε[IA],(?)A∈F,(B2)For any φ∈Cb(R),
Then we restate the Ellsberg-LLN and LLN under E respectively.
Theorem2.4.4Same condition as Theorem2.4.1.Then for any function φ∈Cb(R),
Theorem2.4.5Same conditions as Theorem2.4.2. Given any function φ∈Cb(R),if{Xi}∞i=1is α sequence of φ-convolutionary random variables under E.then
Remark2.4.6If{Xi}∞i=1has higher moments than1,that is E[|Xi|β]<∞for β>1.Note that condition (0.2) for both Theorem2.4.5and Lemma2.3.3holds if supi≥1E[|Xi|β]<∞.Thus by modifying the proof of Lemma3.9in Peng [32],condition φ∈Cb(R)Theorem2.4.5can be extended to the case where φ is continuous with the growth condition φ(x)≤(1+|x|α-1).
(Ⅲ)In Chapter3,we study the application examples of LLN.
Example3.1(Urns Model with ambiguity) Suppose that there exists α countable infinity urns,ordered and indexed by the set N:={1,2,…}.An agent is told that the i-th urn contains100i (100times i) balls with either red or black color,and the number of red balls in the i-th urn is from25i to50i.The agent is told nothing about these urns beyond this information. Only ONE ball will be draun sequently from each urn.As usual,let Xi be the number of red balls for the i-th draw.
Then by Theoren2.4.4, the average number of red balls in n experiments obey the following distribution,
Example3.2(Pricing of a European Option)Let{Bt}t≥0be a Brown-ian motion on a probability space (Ω,F,P).If{St}t≥0is the price of a stock evolving via geometric Brownian: dSt=μStdt+σStdBt. Then pricing a European option in incomplete markets sometimes is trans-mitted to calculate option prices which arranges between upper expectation eμ+σk and lower expectation eμ-σk on its future payoff (St):=(Sr-L)+. Then by Theorem2.4.5, the option price obeys the following rules,
(IV) In Chapter4, we estimate the convergent error of LLNs.
Theorem4.1Let E be a sub-linear expectation and S be its conjugate ex-pectation. Assume that{Xi}∞i=1is a sequence of random variables with finite sup1≤i≤n E[|Xi|2]<∞, then where μ:=|μ|∨|μ|.
When the moment condition is weaken to sup1≤i≤n E[|Xi|1+a]<∞,0<α<1, see Theorem4.2for the corresponding estimations.
众所周知,大数定律(LLN)在概率论与数理统计的发展和应用中发挥了至关重要的奠基作用,与此同时,我们发现有关非可加容度/期望下(强)大数定律的研究成果十分丰富,主要分为两个学术流派：一个是非可加概率流派,其特征是用非可加(不确定的)概率来刻画非可加概率下随机变量的频率属性.另一个是非线性期望流派,其特征是用非可加(上,下)期望来刻画非可加期望下随机变量的频率属性.尽管这两个流派在经典线性概率理论下是等价的,他们在非线性框架下是完全不同的,因为一个非线性期望通常是不能被相应的非线性概率唯一确定的(参见Chen.Z, Chen.T和Davison [7]以及Choquet, Hu, Memin和Peng [12])非可加概率流派成果众多,比如早期经典文献Dow和Verlang [18]以及Walley和Fine [36],近期著名成果例如Cooman和Miranda [15], Epstein和Schneider [19], Marinacci [28], Mac-cheroni和Marinacci [27], Chen和Wu [10], Chen, Wu和Li [11]以及Terdn [34].在对非可加概率的不同假设下,文献证明了当实验次数增加时,通过大量实验得到的实验均值不再逼近于某个确定的期望值,而是在下概率(容度)下落在某个期望值区间内.
在非可加期望流派,Peng是第一个提出g-期望和G-期望的学者.在9-期望的启迪下,Peng[29,30,31]提出了次线性期望下随机变量的独立和同分布定义,我们称之为Peng独立.在某些对非可加期望的假设下,Peng通过偏微分方程(PDE)理论给出了次线性期望下的大数定律和中心极限定理.
将非可加概率和非可加期望两个学派的成果与经典大数定律相比较,我们发现,在对概率和期望的公理化性质进行削弱的同时,我们必须对状态空间,非可加概率,随机变量做出额外的技术性假设作为补偿.
自然地,我们想到如下问题：可否借鉴经典可加概率(Feller,Linderberg等人)的证明方法,从概率的角度将大数定律推广到次线性期望下呢?答案是肯定的.本文中,我们首先借鉴了经典的Linderberg-Feller的证明思想在事件独立下给出了Choquet期望下的大数定理,然后推广到更一般情形：卷积独立下的次线性期望下大数定律,进而获得容度下的弱大数定律,并给出了一个与二者相关的等价定理.本文证明过程仅使用了泰勒展开式,次线性期望性质等纯概率基础工具,未采用特征函数或PDEs等复杂辅助手段.进一步,与文献相比,我们削弱了大数定理的假设条件.比如,在次线性期望下,降低了随机变量的阶矩条件,随机变量独立性假设也弱化为卷积独立.此外,我们的次线性大数定律能够涵盖著名的Ellsberg模型(带有模糊性的罐子模型),而Ellsberg模型因为具有概率模糊性并不符合文献中大数定律的假设条件.
本文共分为四章.第一章研究在事件独立定义下,Choquet期望下的大数定律.第二章研究在卷积独立定义下,一般次线性期望下的大数定律.第三章给出次线性大数定律在模糊条件下的应用.第四章给出了次线性大数定律的收敛误差估计.
引入符号：
假设Ω为状态空间,F为σ-域.称函数X:Ω→R为可测空间(Ω,F)上随机变量,若X是F-可测的.令H为可测空间(Ω,F)上随机变量全体的子集.
假设Gb(R)为R上所有有界连续函数的集合C+b(R)为Cb(R)上的非负单调函数的全体C2b(R)为Cb(R)上那些函数一阶,二阶导数都存在且导数仍在Gb(R)中的函数全体.对给定有限常数μ和μ,记集合Dn:={y|y=(y1,y2,…,yn),yi∈[μ,μ],1≤i≤n}.
(Ⅰ)第一章主要研究上Choquet期望下的大数定律.
容度和Choquet期望在定义形式,基本性质等方面与概率和线性期望具有高度相似性.可以说,Choquet理论是联系经典线性理论与新兴次线性理论的桥梁.1999年到2005年,Maccherroni和Marinacci [27][28]提出了容度下的随机变量独立性定义,独立形式与概率下的独立定义相似.正是考虑到Choquet期望与线性期望的这种相似性,在将大数定律向非可加期望领域推广时,我们优先选择Choquet期望作为尝试的起点.所以本文先讨论Choquet这种相对简单的情况(此时类似线性期望有较多借鉴之处),在事件独立的假设下,给出了Choquet期望下大数定律.然后再考虑一般的次线性期望下大数定律.从简入难.
回顾Choquet期望下大数定律的理论进展,我们发现在事件独立的假设下,不同文献给出了不同技术假设,但从证明方式来看,主要分为以下两种：一种是转化法：“曲线救国”,比如Chareka [3]将不可加的Choquet积分转化成可加的Lebesgue-Stieltjes积分.然后利用Lebesgue-Stieltjes积分性质证明了Choquet框架下的强(弱)大数定律.另一种是直接证明法：比如Li和Chen[26]直接证得容度下的Chebyshev不等式和Borel-Cantelli引理,从而证明了Choquet期望下大数定律,证明方法类似线性LLN.由此启发我们：在事件独立下,可否将大数定律的其他(Linderberg, Feller等)经典证法推广到Choquet期望下证得大数定律呢?我们的答案是肯定.
我们知道,证明大数定律的关键条件是概率/期望的可加性和随机变量的阶矩条件.而本章讨论的Choquet期望恰为非可加期望.为了解决这个期望非可加问题,我们采用2-alternating容度,因为由2-alternating容度生成的Choquet期望具有我们所需的次可加性.在这个前提假设下,本章讨论了独立同分布随机变量序歹(?){Xi}∞i=1的依分布收敛(分布极限)问题.此外,对比其他Choquet结论,我们的Choquet大数定律对随机变量的阶矩条件也进行了弱化.
在引入大数定律前,我们先叙述三个核心引理作为铺垫.和变通项引理：
引理1.3.1令V为F上的2-alternating容度,且Cv,Cv分别为由其生成的上,下Choquet期望.令{Xi}i=1∞为(Ω,F)上的一列独立随机变量.则对任意单调函数φ∈Gb(R)和任意常数yi∈R,其中n
泰勒展开引理：引理1.3.2令V为2-alternating容度,Cv,Cv分别为其生成的上,下Choquet期望令{Xi)i=1∞为同分布随机变量列且满足Cv[Xi]=μ和Cv[Xi]=μ假设对任意i≥1,有Cv[|Xi|]<∞.则对任意函数φ∈Cb2(R),存在一个正值常数b。(∈)使得bn((?))→0当n→∞时,从而(Ⅰ)∑i=1n supx∈R{Cv[φ(x+xi/n)]-φ(x)≤supx∈R G(φ'(x),μ,μ)+bn((?)).(Ⅱ)∑i=1n infx∈R{Cv[φ(x+Xi/n)]-φ(x)≥infx∈R G(φ'(x),μ,μ)-bn((?)).其中G(x,μ,μ):=x+μ-x-μ.
引理1.3.3令G(x,y,z)函数定义同引理1.3.2,即G(x,y,z):=x+y-x-z.则对单调的φ∈Cb(R),有(Ⅰ)infy∈Dn supx∈R G(φ'(x),μ-1/n∑ni=1yi,μ-1/n∑i=1n yi)=0.(Ⅱ)infy∈Dn infx∈R G(φ'(x),μ-1/n∑ni=1yi,μ-1/n∑i=1n yi)=0.
经过上述三个引理的铺垫,我们可以引入本章第一个定理：分布极限定理.此定理表明,在Choquet期望下实验均值的分布极限是一个最大分布.
定理1.4.1(分布极限定理)令V为F上2-alternating容度,Cv,Cv分别为由其生成的上,下Choquet期望.假设{Xi}∞i=1为独立同分布随机变量列且满足Cv[Xi]=μ,Cv[Xi]=μ记部分和Sn:=∑ni=1Xi假设对任意i≥1,Cv[Xi]<∞.则对任意单调函数φ∈Cb(R),
定理1.4.2(容度下的弱大数定律)令V为F上2-alternating容度,Cv,Cv分别为由其生成的上,下Choquet期望.令v(A):=Cv[IA],(?)4∈F假设{Xi}∞i=1为独立同分布随机变量列且满足Cv[Xi]=μ,Cv[Xi]=μ.记Sn:=∑ni=1Xi假设对任意i>1,Cv[|Xi|]<∞若对任意φ∈Cb+(R),任意∈>0,则有
下面,我们给出一个令最大分布等价于容度下弱大数定律的充分条件.
定理1.4.3(等价定理)令V为F上2-alternating容度,Cv,Cv分别为由其生成的上,下Choquet期望.给定函数φ∈Cb+(R),假设{Xi}∞不=1为一列独立同分布随机变量列并满足Cv[Xi]=μ,Cv[Xi]=μ假设对任意i>1,Cv[|Xi|]<∞.令Sn:=∑ni=1Xi则结论(A)与(B)等价.(A)对任意(?)>0,令v(4):=Cv[IA],VA∈F,有(?)v(μ-ε≤Sn/n≤μ+ε)=1.(B)对任意φ∈Cb(R),
等价定理的意义在于：若收敛结论对单调φ∈Cb(R)成立,则对任意φ∈Gb(R)都成立.我们由分布极限定理推广得到如下Choquet期望下大数定律.
定理1.4.4(Choquet期望下大数定律)令V为F上2-alternating容度,Cv,Cv分别为由其生成的上,下Choquet期望.假设{Xi}∞i=1为独立同分布随机变量列且满足Cv[Xi]=μ,Cv[Xi]=μ假设对任意i≥1,Cv[|Xi|]<∞.记Sn:=∑ni=1,Xi则对任意函数φ∈Cb(R),有
注1.4.5由定理证明过程可知,随机变量同分布条件可以削弱为:随机变量具有有限共一阶矩,即{Xi}∞i=1满足1≤i≤n, Cv[Xi]=Cv[Xi], Cv[Xi]=Cy[X1]； Cv[|XI|]=Cv[|x1|], Cv[|Xi|]=Cv[|X1|]<∞.
(Ⅱ)在第二章,我们用概率语言证得了次线性期望下的大数定律.
本章是经典大数定律的自然推广.本章有四个主要结论：(1)我们对广义Ells-berg模型的极限分布进行了研究,并发现它的极限分布是一个最大分布.(2)我们将Ellsberg模型推广,得到了一个有关随机变量的充分条件,在这个条件下,实验均值的极限分布与Ellsberg模型的极限分布是一致的.(3)在次线性期望的φ-卷积独立定义下,我们给出了最大分布等价于容度下弱大数定律的充分条件.(4)我们将本章结论与文献结论进行了对比.在线性期望下卷积独立的启发下,我们将卷积独立这个概念推广到次线性期望之下.和变通项引理：
引理2.3.1给定函数φ∈Cb(R)假设E为次线性期望,ε为其共轭期望.令{Xi}i=1∞为E下一列φ-卷积独立随机变量.则对任意常数yi∈R,1≤i≤n,有其中n则上述引理有如下变形：引理2.3.2令P为概率测度集,{Xi)i=1∞在每一个概率Q∈P下都是一列独立随机变量.则对任意常数yi∈R,i=1,2,…,n,和任意函数φ∈Cb(R),有其中
下述泰勒展开引理是证明Choquet大数定律时的核心引理.我们将(0.1)式视为次线性期望E下的Linderberg条件.
引理2.3.3假设E为次线性期望,￡为其共轭期望.假设随机变量序列{Xi}i=1∞具有有限共一阶矩且E[Xi]=μ,ε[Xi]=μ假设对任意∈>0,有则对任意单调函数φ∈C2b(R),(Ⅲ)特别的,若E[·]和ε[·]为概率测度集P上的上,下期望算子且满足
则对任意单调函数φ∈C2b(R),经过上述引理的铺垫,我们引出容度下/次线性期望下的大数定律.
定理2.4.1(Ellsberg型大数定律)给定一个概率测度集P,令(E,ε)分别为P上EQ生成的上,下期望.假设对任意Q∈P,{Xi}∞i=1是Q下一列独立随机变量,{Xi}i=1n具有有限共一阶矩(?)μ:=E[Xi],μ:=ε[Xi]使得假设条件(0.1)成立.记Sn:=∑ni=1Xi进一步,若则(Ⅰ)对任意单调函数φ∈Cb(R),我们有F,则对任意∈>0,
定理2.4.2(次线性期望下的大数定律)假设E是一个次线性期望,￡是一个共轭期望.假设随机变量序列{Xi}∞i=1具有有限共一阶矩且E[Xi]=μ和ε[Xi]=μ.假设条件(0.1)成立.记部分和Sn:=∑ni=1Xi则有
(Ⅰ)给定单调函数φ∈Cb(R),若{Xi}∞i=1是E下一列φ-卷积独立随机变量,则
(Ⅱ)若对任意φ∈C+b(R),{X}∞i=1是E下一列φ-卷积独立随机变量,令v(A)：=ε[IA],VA∈(?),则对任意∈>0,
类似第一章Choquet期望的结构,Ellsberg型大数定律(定理2.4.1)和次线性期望下大数定律(定理2.4.2)中的函数φ都局限于Cb(R)上的单调函数,为此我们引入次线性期望下的等价定理(定理2.4.3),将对单调φ∈Cb(R)成立的定理推广到对任意φ∈Cb(R)成立.
定理2.4.3(次线性期望下的等价定理)假设E是一个次线性期望,ε是它共轭期望.对函数φ∈C+b(R),假设{Xi}∞i=1是一列φ-卷积独立的随机变量,具有有限共一阶矩且E[Xi]=μ和ε[Xi]=μ使得假设条件(0.1)成立.令Sn：=∑ni=1Xi则结论(A2)与(B2)等价.
似(A2)对任意∈>0,(B2)对任意φ∈Cb(R),
通过等价定理可知,Ellsberg型/E下的大数定律(Ⅰ)对任意φ∈Cb(R)都成立.表述如下.
定理2.4.4假设条件同定理2.4..1.则对任意函数φ∈Cb(R),有
定理2.4.5假设条件同定理2.4.2.给定任意φ∈Cb(R),若{Xi}∞i=1.是E下一列φ-卷积独立随机变量,则
注2.4.6若{Xi)∞i=1的阶矩阶数大于1,即对任意常数β>1,E[|Xi|β<∞.注意到如果supi≥1E[|Xi|β]<∞,则定理2.4.5和引理2.3.3中的假设条件(0.1)都成立.所以根据Peng [32]中引理3.9的证明,定理2.4.5的条件φ∈Cb(R)就可以弱化为:连续函数φ满足增长条件|φ(x)|≤C(1+|x|β-1),去掉有界性.
(Ⅲ)第三章,次线性大数定律在模糊条件下的应用.
例子3.1(带有模糊性的罐子模型)考虑有限可数个罐子,按顺序将编号记为{1,2,…}.实验者被告知第i个罐子里有100i个小球(此后100i表示100乘以i),颜色为红色或者黑色.第i个罐子里红色球的个数为25i到50i个不等.实验者不知道除此之外的任何信息.每一次只能从一个罐子里取出一个小球.记由定理2.4.4知:当实验次数足够多,n次试验中摸到红球的个数的实验均值服从如下最大分布
例子3.2(期权定价模型)令{B)t≥0为概率空间(Ω,F,P)上的几何布朗运动.{St≥o是服从几何布朗运动的股票价格:dSt=μStdt+σStdBt.在非完全市场下,欧式期权的未来损益函数为φ(ST):=(ST-L)+因为上期望是eμ+σκ,下期望是eμ-σκ,由定理2.4.5知股价的分布极限如下
(Ⅳ)第四章,主要研究次线性大数定律的收敛误差估计.
定理4.1假设E是一个次线性期望,￡是一个共轭期望.假设随机变量序列{Xi)∞i=1具有有限共一阶矩,E[Xi]=-μ,ε[Xi]=μ.若记部分和Sn:=∑ni=1Xi则二阶矩下的大数定律收敛误差估计如下.其中μ:=|μ|∨|μ|.
当阶矩条件降低至(?)supl≤i≤n E[|Xi|1+α]<∞,0<α<1,误差估计相关结果见定理4.2.
Ever since the definition of capacity was introduced by Choquet [13], it has been a heated scientific subject worldwide. Since in many applica-tion fields, such as finance, economics and robust statistics, the traditional additive probability measures fail to provide adequate or good information to describe or interpret the uncertain phenomena accurately. Therefore, in some areas, the assumption that given a precise probability for random vari-ables as an apparently quite natural property has been abandoned, in favor of non-additive/imprecise probabilities (such as lower-upper probabilities) or nonlinear expectations (for example Choquet expectations, sublinear expec-tations, lower-upper expectations). Indeed, as early as1954, Keynes [24] had discovered this problem, then constructed a theory of imprecise prob-ability. Meanwhile, capacities, the non-additive probability measures, seem to be a powerful tool to model the uncertainty when the assumption of addi-tivity is suspect.(e.g., Augustin [1], Maccherroni and Marinacci [27], Doob [17], Schmeidler [33]). Later, motivated by mathematical finance and ro-bust statistics, the frequentist properties of random variables for non-additive (lower and upper) probabilities have attracted more and more attentions.
As we known, the law of large numbers (LLN) plays an important role in probability theory and mathematical statistics. Meanwhile, in the literature, various generalizations of (strong) LLNs for non-additive probabilities have been established. Among them turn into two groups:one is called nonad-ditive probability group in which people like to use nonadditive (imprecise) probability to accommodate the frequentist properties of random variables for non-additive probability, the other is called nonlinear expectations group in which people like to use nonadditive (lower and upper) expectations to accommodate the frequentist properties of random variables for non-additive expectations. Although both groups in the framework of linear probability theory are equivalent, they are totally different in the nonlinear case in the sense that a nonlinear expectation usually could not be determined uniquely by the corresponding nonlinear probability (see for example,[7],[12]). In nonadditive probability group, the earlier papers by Dow and Werlang [18] and Walley and Fine [36], while the more recent results by Cooman and Miranda [15], Epstein and Schneider [19], Marinacci [28], Maccheroni and Marinacci [27], Chen and Wu [10], Chen, Wu and Li [11] and Teran [34]. Under various assumptions on nonadditive probabilities, they proved that the frequentist obtained from a large number of trials is no longer close to an expected value but an interval of possible expected value under lower probability as more trials are performed.
In nonadditive expectation group, Peng is the first to introduce the notions g-expectation and G-expectation. Inspired by g-expectation, Peng [29,30,31] introduces a new notion of identical and independently distributed (ⅡD) random variables under sub-linear expectations, which is called Peng independence. Under some assumptions on nonadditive expectations, he shows LLNs by using partial differential equations (PDEs).
Comparing the results obtained by the two groups with the classical LLN, one could find that the weakening of axiomatic properties of probability and expectation has been balanced by the incorporation of the extra technical assumptions on the state space and/or nonadditive probabilities and/or random variables.
A natural question is arising:Can we extend the LLN to the sub-linear expectations case by traditional pure probabilistic (Linderberg-Feller's) method, without using the characteristic functions or PDEs? The answer is affirmative. In this paper, we first extend the classical LLN to the Choquet expectation case with event independence under capacity. Then we extend LLN to the sub-linear expectation case with convolutionary random vari- ables. In both cases, we establish the equivalence theorem to connect the "distribution theorem" with the "Weak LLN under capacity". The whole proving method we proposed here only depends on some elementary prob-abilistic techniques, such as Taylor expansions and the basic properties of sub-linear expectations, without using any artificial tools like characteristic functions or PDEs. In this sense, our LLN is a natural extension of the classical LLN. Further, compared to other literature, our non-additive ver-sion of LLN has weaken the assumptions of the theorems, for example, we weaken the moment condition of the random variables, as well as replace the independence condition by the weaker notion of φ-convolution on ran-dom variables. Meanwhile, Ellsberg model satisfies our theorem assumptions.
This paper is divided into4chapters. In chapter1, under the event inde-pendence for capacity, we establish law of large numbers for Choquet expec-tations induced by2-alternating capacities. In Chapter2, we first introduce the notion of convolutionary random variables under sub-linear expectations, then prove the LLNs under both sub-linear expectations and capacities. In Chapter3, we give three application examples of our LLNs, especially the Ellsberg model(Urn model with ambiguity). The corresponding estimation of the convergence error of our LLNs is given in Chapter4.
Notations: Suppose that Ω is a state space and (?) is a σ-field. Function X:Ω→R is called a random variable on measurable space (Ω,(?)), if X is F-measurable. Let H be a subset of all random variables on measurable space (Ω,F).
Suppose that Cb(R) is the set of all bounded and continuous real-valued functions on R and Ckb(R) is the set of bounded and k-time continuously differentiable functions with bounded derivatives of all orders less than or equal to k. C+b(R) is all the non-negative monotonic functions in Cb(R)
For given finite constants μ and μ, set Dn:={y:=(y1,y2,…,yn):yi∈[μ,μ], i=1,2,…,n}.
(I) In Chapter1, we study the law of large numbers under Choquet expectations.
As we known, the definitions and properties of capacity/Choquet ex-pectations are quite similar to those of probability/linear expectations, thus, the Choquet theory can be viewed as a bridge well connecting the traditional probability theory with the new-arising capacity theory. From1999to2005, Maccherroni and Marinacci [27][28] introduce event independence under ca-pacity, which is in accord with the probability case. Due to this similarity, we extend the classical LLN starting from the Choquet expectation case, then consider the more general and complex case:the sub-linear expectation case.
Reviewing the literature of LLN under Choquet expectations, different authors propose different assumptions of LLN. However, the main pattern of proving LLN can be divided into2modes:one is the indirect method, such as Chareka [3], he turned the non-additive Choquet integral into the additive Lebesgue-Stieltjes integral, then by using the existing properties of Lebesgue-Stieltjes integral he derived the LLN under Choquet expectations. The other is the direct method, such as Li and Chen [26], they obtained LLN by proving Chebyshev inequality and Borel-Cantelli lemma under capacity, still following the proving pattern of traditional LLN. These inspire us with a rising question:With event independence under capacity, can we extend the LLN to Choquet expectations by pure probabilistic (Linderberg-Feller's) way? The answer is affirmative.
Recall that the key to prove classical LLN are the additivity of proba-bilities/expectations as well as the moment conditions of random variables. However, the Choquet expectations happen to be the non-additive ones. To overcome this non-additive problem, we adopt the2-alternating capacity, thus the Choquet expectations induced by it turn to be sub-additive ones, under which we consider the sequence{Xi}∞i=1of independent and identically distributed (ⅡD) random variables converges in distribution. Moreover, our moment condition on random variables is weaker than other literature.
Lemma1.3.1Let V be a2-alternating capacity defined on (?),and Cv,Cv be the induced upper,lower Choquet expectation respectively.Let {Xi}i=1∞be a sequence of independent random variables.Then for any monotonic φ∈Cb(R) and any constant yi∈R,
Lemma1.3.2Let V be a2-alternating capacity and Cv,Cv be the induced upper,lower Choquet expectation respectively.Let {Xi}i=1∞be a sequence of identical distributed random variables with Cv[Xi]=μ and Cv[Xi]=μ satisfying that for i≥1, Cv[|Xi|]<∞.
Then,for each function φ∈Cb2(R),there exists a positive constant bn(∈)with bn(∈)→0,as n→∞,such that (Ⅰ)∑i=1n supx∈R{Cv[φ(x+Xi/n)]-φ(x))≤supx∈R G(φ'(x),μ,μ)+bn(∈).(Ⅱ)∑i=1n,infx∈R{Cv[φ(x+Xi/n)]-φ(x)}≥infx∈R G(φ'(x),μ,μ)-bn(∈). Where G(x,μ,μ):=x+μ-x-μ.
Lemma1.3.3Let G(x,y,z) be the function defined in Lemma1.3.2,that is G(x,y,z):=x+y-x-z. Then for any monotonic φ∈Cb(R),
(Ⅰ)infy∈Dn supx∈R G(φ'(x),μ-1/n∑i=1n yi,μ-1/n∑i=1n yi)=0. (Ⅱ)infy∈Dn infx∈R G(φ'(x),μ-1/n∑ni=1yi,μ-1/n∑ni=1yi)=0.
Our new LLNs under Choquet expectations are stated as follows. Theorem1.4.1(Limit Distribution Theorem)Let V be a2-alternating capacity defined on (?),and Cv,Cv be the induced upper, lower Choquet expectation respectively.Let {Xi}∞i=1be a sequence of ⅡD random variables on (Ω,(?))with Cv[Xi]=μ,Cv[Xi]=μ.ASSume that for i≥1, Cv[|Xi|]<∞Set Sn:=∑ni=1Xi. Then for each monotonic function φ∈Cb(R),(Ⅰ)(?)Cv[φ(Sn/n)]=supμ≤x≤μ φ(x)；(Ⅱ)(?)Cv[φ(Sn/n)]=infμ≤x≤μ φ(x).
The limit distribution theorem indicates that the limit distribution of the empirical average of random variables is the maximal distribution.
Theorem1.4.2(Weak LLN under capacity)Let V be a2-alternating capacity defined on (?),and Cv,Cv be the induced upper,lower Choquet ex-pectation respectively.Let v(A):=Cv[IA],(?)A∈(?).Let{Xi}∞i=1be a sequence of ⅡD random variables with Cv[Xi]=μ,Cv[Xi]=μ.Assume that for i≥1, Cv[|Xi|]<∞. Set Sn:=∑ni=1Xi. If for any function φ∈C+b(R),any (?)>0,then
Theorem1.4.3(Equivalence Theorem)Let V be a2-alternating capacity defined on (?),and Cv,Cv be the induced upper,lower Choquet expectation respectively.Given function φ∈C+b(R),Let{Xi)∞i=1be a sequence of ⅡD random variables with Cv[Xi]=μ,Cv[Xi]=μ.Assume that for i≥1, Cv[|Xi|]<∞. Let Sn:=∑ni=1Xi.Then the followings are equivalent. (A) For any (?)>0, let v(A):=Cv[IA],(?)A∈F, then (B) For any φ∈Cb(R),
The equivalence theorem states that, if the convergent result is true for any monotonic function φ∈Cb(R), then it is still true for any φ∈Cb(R)
Theorem1.4.4(LLN under Choquet expectations) Let V be a2-alternating capacity defined on (?), and Cv, Cv be the induced upper, lower Choquet expectation respectively. Let{Xi}∞i=1be a sequence of IID random variables with Cv[Xi]=μ, Cv[Xi]=μ. Assume that for i≥1, CV[|Xi|]<∞. Set Sn:=∑ni=1Xi.Then for each function φ∈Cb(R),
Remark1.4.5Further, the condition of identical distribution in above the-orems can be weaken to "finite common first moment condition", that is, for1≤i≤n, Cv[Xi]=Cv[X1], Cv[Xi]=Cv[X1]; Cv[|Xi|]=Cv[|X1|], Cv[|Xi|]=Cv[|X1|]<∞.
(Ⅱ) In Chapter2, we prove the LLNs under sub-linear ex-pectations by using pure probabilistic method, without using the characteristic functions or PDEs. It is a nature extension of the tradition LLNs.
We obtain four principal results in this chapter.(1) We explore the limit distribution of general Ellsberg-type model mentioned above and show that its limit distribution is a maximal distribution.(2) We extend Ellsberg-type model to more general case, and obtain a sufficient condition on ran-dom variables and sub-linear expectations, under which the empirical average of random variables has the same limit distribution as Ellsberg-type model does.(3) With a new notion of φ-convolution on sub-linear expectations and random variables, we show both maximal distribution and weak LLNs are equivalent.(4) We compare our results with which appearing in the men-tioned articles.
Lemma2.3.1Let E be a sub-linear expectation and ε be its conjugate ex-pectation. Given a function φ∈Cb(R). Let{Xi}∞i=1be a sequence of φ-convolutionary random variables under E. Then for any constant yi∈R, i=1,2,…,n,
If sub-linear E is the upper expectation operator generated by a set P of probability measures, that is E[·]:=(?)Eq[·], then Lemma2.3.1can be restated as follows.
Lemma2.3.2Let P be a set of probability measures and{Xi}∞I=1be a sequence of independent random variables for each Q∈P. EQ is the linear expectation with respect to probability Q. Then for any constant yi∈R., i=1,2,…,n,and any function φ∈Cb(R), where
The following lemma is a key to the proof of our LLN.Condition(0.2) can be viewed as the Linderberg condition under sub-linear expectation E.
Lemma2.3.3Let E be a sub-linear expectation and ε be its conjugate ex-pectation.Suppose that{Xi}i=1∞is a sequence of random variables with finite common first moment and E[Xi]=μ,ε[Xi]=μ,i≥1.Moreover,for any (?)>0,if then for any monotonic function φ∈Cb2(R),we have (Ⅰ)limn→∞infy∈Dn∑i=1n supx∈R{E[φ(x+Xi-yi/n)]-φ(x)}=0；(Ⅱ)limn-→∞infy∈Dn∑i=1n infx∈R{E[φ(x+Xi-yi/n)]-φ(x))=0.(Ⅲ)Moreover,if E[·] and ε[·] are the upper and lower expectations on the
set P of probability measures such that
then,for any monotonic function φ∈Cb2(R),
We are now ready to prove the LLN for sub-linear expectations and ca-pacities. Similar to the structure of Chapter1, We first present the Ellsberg-LLN(Theorem2.4.1),then the sublinear LLN(Theorem2.4.2).
Theorem2.4.1(Ellsberg-LLN) Given a set P of probability measures, let E, ε be the upper and lower expectations of EQ over P respectively. Assume that for any Q∈P,{Xi}∞i=1is a sequence of independent random variables under Q, and{Xi}ni=1has the finite common first moment μ:=E[Xi] and μ:=ε[Xi] such that condition (0.2) holds. Set Sn:=∑ni=1Xi.Furthermore, if (Ⅰ) For any monotonic φ∈Cb(R), then (Ⅱ)For φ∈C+b(R). Set V(A):=(?)Q(A), v(A):=(?)Q(A),(?)A∈F, then for any (?)>0,
Next, we extend the Ellsberg-LLN with event independence to the LLN under sub-linear expectation E with φ-convolutionary random variables. It is worthwhile to note that the limit distribution of the empirical average under E is still a maximal distribution.
Theorem2.4.2(LLN for sub-linear expectations) Let E be a sub-linear expectation and ε be its conjugate expectation. Assume that{Xi}∞i=1is a sequence of random variables with finite common first moment-E[Xi]=μ
(Ⅰ)Given monotonic φ∈Cb(R),if {Xi}∞i=1is a sequence of φ-convolutionary random varables under E,then
(Ⅱ)If for any φ∈Cb+b(R),{Xi}∞i=1is a sequence of φ-convolutionary randorn variables under E,then for any (?)>0,and for v(A):=ε[IA],(?)A∈F, we have
Since the function φ in both LLNs is limited to the monotonic φ∈Cb(R),we need the following equivalence theorem to extend the monotonic φ∈Cb(R) to all φ∈Cb(R).
Theorem2.4.3(Equivalence Theorem under E)Let E be a sub-linear expectation and εbe its conjugate expectation.Given function φ∈Cb+(R), assume that {Xi}∞i=1is a sequence of φ-convolutionary random variables with finite common first moment E[Xi]=μ and ε[Xi]=μ.If condition (0,2)(A2) For any (?)>0,v(A):=ε[IA],(?)A∈F,(B2)For any φ∈Cb(R),
Then we restate the Ellsberg-LLN and LLN under E respectively.
Theorem2.4.4Same condition as Theorem2.4.1.Then for any function φ∈Cb(R),
Theorem2.4.5Same conditions as Theorem2.4.2. Given any function φ∈Cb(R),if{Xi}∞i=1is α sequence of φ-convolutionary random variables under E.then
Remark2.4.6If{Xi}∞i=1has higher moments than1,that is E[|Xi|β]<∞for β>1.Note that condition (0.2) for both Theorem2.4.5and Lemma2.3.3holds if supi≥1E[|Xi|β]<∞.Thus by modifying the proof of Lemma3.9in Peng [32],condition φ∈Cb(R)Theorem2.4.5can be extended to the case where φ is continuous with the growth condition φ(x)≤(1+|x|α-1).
(Ⅲ)In Chapter3,we study the application examples of LLN.
Example3.1(Urns Model with ambiguity) Suppose that there exists α countable infinity urns,ordered and indexed by the set N:={1,2,…}.An agent is told that the i-th urn contains100i (100times i) balls with either red or black color,and the number of red balls in the i-th urn is from25i to50i.The agent is told nothing about these urns beyond this information. Only ONE ball will be draun sequently from each urn.As usual,let Xi be the number of red balls for the i-th draw.
Then by Theoren2.4.4, the average number of red balls in n experiments obey the following distribution,
Example3.2(Pricing of a European Option)Let{Bt}t≥0be a Brown-ian motion on a probability space (Ω,F,P).If{St}t≥0is the price of a stock evolving via geometric Brownian: dSt=μStdt+σStdBt. Then pricing a European option in incomplete markets sometimes is trans-mitted to calculate option prices which arranges between upper expectation eμ+σk and lower expectation eμ-σk on its future payoff (St):=(Sr-L)+. Then by Theorem2.4.5, the option price obeys the following rules,
(IV) In Chapter4, we estimate the convergent error of LLNs.
Theorem4.1Let E be a sub-linear expectation and S be its conjugate ex-pectation. Assume that{Xi}∞i=1is a sequence of random variables with finite sup1≤i≤n E[|Xi|2]<∞, then where μ:=|μ|∨|μ|.
When the moment condition is weaken to sup1≤i≤n E[|Xi|1+a]<∞,0<α<1, see Theorem4.2for the corresponding estimations.
引文
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[3]Chareka, P., (2009). The central limit theorem for capacity. Stat. and Prob. Letters,79,1456-1462.
[4]Chateauneuf, A. and Lefort, J. P. (2008). Some Fubini theorems on product σ-algebras for non-additive measures. Internat. J. Approx. Reason.48686-696.
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[7]Chen, Z., Chen, T., and Davison, M. (2005). Choquet expectations and Peng's g-expectations. Ann. Prob.33 1179-1199.
[8]Chen, Z., and Epstein, L. G. (2002). Ambiguity, risk and asset returns in continuous time. Econometrica 70 1403-1443.
[9]Chen, Z., (2010). The strong law of large numbers for capacities. arX-iv:1006.0749 [math.PR].
[10]Chen, Z., and Wu, P. (2011). Strong laws of large numbers for bernoulli experiments under ambiguity. Adv. Intell. Soft Comput.10019-30.
[11]Chen, Z., Wu, P., and Li, B (2013). A strong law of large numbers for non-additive probabilities. Internat. J. Approx. Reason.54 365-377.
[12]Choquet. F.. Hu, Y., Memin, J., and Peng, S. (2002). Filtration-consistent nonlinear expectations and related g-expectations. Prob. Theory and Relat-ed Fields 123 1-27.
[13]Choquet, G., (1954) Theory of capacities, Annales de l'institut Fourier,5, 131-295.
[14]Chung, K.L., (1974) A course in probability theory,2nd edn., Academic Press, New York.
[15]Cooman, D., and Miranda, E. (2008). Weak and strong laws of large numbers for coherent lower previsions. J. Statist. Plan. Inference 138 2409-2432.
[16]Dennerberg, D.,(1994). Non-additive measure and intergral, Kluwer, Dor-drecht.
[17]Doob, J., (1984) Classical potential theory and its probabilistic counterpart, Springer, New York.
[18]Dow, J., and Werlang, S. (1994). Learning under Knightian uncertainy:the laws of large numbers for non-additive probabilities. Unpublished.
[19]Epstein, L. G. and Schneider, M. (2003). ⅡD:independently and indisting-guishably distributed. J. Econ. Theory 113 32-50.
[20]Feller, W. (1971). An introduction to probability theory and its applications. Vol. II,2nd ed. John Wiley & Sons, Inc., New York.
[21]Ghirardato, P. (1997). On independence for non-additive measures, with a Fubini theorem. J. Econom. Theory 73 261-291.
[22]HUBER, P. J. (1973). The use of Choquet capacities in statistics. Bull. Inst. Internat. Statist.45 181-191.
[23]HUBER, P. J. and RONCHETTI, E. M. (2009). Robust Statistics,2nd ed. Wiley, New Jersey.
[24]KEYNES, J. M. (1921). A treatise on probability. Macmillan, London.
[25]Le CAM, L. (1986). The central limit theorem around 1935. Statist. Sci.1 78-96.
[26]Li, W. and Chen Z.,(2011). Law of large numbers of negatively correlated random variables for capacity, Acta Mathematicae Applicatae Sinica, En-glish Series,27(4),749-760.
[27]Maccheroni, F., and Marinacci, M., (2005). The strong law of large numbers for capacities, The Annals of Probability,33 (3),1171-1178.
[28]Marinacci, M.,(1999) Limit laws for non-additive probabilities and their fre-quentist interpretation, Journal of Economic Theory,84,145-195.
[29]Peng, S.,(2007). Law of large number and central limit theorem under non-linear expectations, arXiv:0702358vl [math.PR].
[30]Peng, S., (2008). A new central limit theorem under sublinear expectations, arXiv:0803.2656vl [math.PR].
[31]Peng, S., (2009). Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expec-tations, Science in China Series A-Mathematics,52(7),1391-1411.
[32]PENG, S. (2010). Nonlinear expectations and stochastic calculus under un-certainy. arXiv preprint arXiv:1002.4546.
[33]Schmeidler, D., (1989). Subjective probability and expected utility without additivity, Econometrica,57,571-587.
[34]Teran, P. (2013). Laws of large numbers without additivity. Trans. Amer. Math. Soc. Accepted.
[35]Shapley, L., (1971). Cores of convex games, International Journal of Game Theory,1,11-26.
[36]Walley, P., and Fine, T. L. (1982). Towards a frequentist theory of upper and lower probability. Ann. Statist.10 741-761.
[37]Wasserman, L. A. and Kadane, J., (2009). Bayes'theorem for Choquet ca-pacites. Annal of Statistics,18,1328-1339.
[2]BOREL, E. (1962). Probabilities and Life. Translated from the French by Maurice Baudin Dover Publications, Inc., New York.
[3]Chareka, P., (2009). The central limit theorem for capacity. Stat. and Prob. Letters,79,1456-1462.
[4]Chateauneuf, A. and Lefort, J. P. (2008). Some Fubini theorems on product σ-algebras for non-additive measures. Internat. J. Approx. Reason.48686-696.
[5]Chen, J. and Chen, Z.,(2014). The generalized law of large numbers under Choquet expectations.
[6]Chen, Z. and Chen, J.,(2013). Limit theorems for sub-linear expectations. Anna. Prob. preprint.
[7]Chen, Z., Chen, T., and Davison, M. (2005). Choquet expectations and Peng's g-expectations. Ann. Prob.33 1179-1199.
[8]Chen, Z., and Epstein, L. G. (2002). Ambiguity, risk and asset returns in continuous time. Econometrica 70 1403-1443.
[9]Chen, Z., (2010). The strong law of large numbers for capacities. arX-iv:1006.0749 [math.PR].
[10]Chen, Z., and Wu, P. (2011). Strong laws of large numbers for bernoulli experiments under ambiguity. Adv. Intell. Soft Comput.10019-30.
[11]Chen, Z., Wu, P., and Li, B (2013). A strong law of large numbers for non-additive probabilities. Internat. J. Approx. Reason.54 365-377.
[12]Choquet. F.. Hu, Y., Memin, J., and Peng, S. (2002). Filtration-consistent nonlinear expectations and related g-expectations. Prob. Theory and Relat-ed Fields 123 1-27.
[13]Choquet, G., (1954) Theory of capacities, Annales de l'institut Fourier,5, 131-295.
[14]Chung, K.L., (1974) A course in probability theory,2nd edn., Academic Press, New York.
[15]Cooman, D., and Miranda, E. (2008). Weak and strong laws of large numbers for coherent lower previsions. J. Statist. Plan. Inference 138 2409-2432.
[16]Dennerberg, D.,(1994). Non-additive measure and intergral, Kluwer, Dor-drecht.
[17]Doob, J., (1984) Classical potential theory and its probabilistic counterpart, Springer, New York.
[18]Dow, J., and Werlang, S. (1994). Learning under Knightian uncertainy:the laws of large numbers for non-additive probabilities. Unpublished.
[19]Epstein, L. G. and Schneider, M. (2003). ⅡD:independently and indisting-guishably distributed. J. Econ. Theory 113 32-50.
[20]Feller, W. (1971). An introduction to probability theory and its applications. Vol. II,2nd ed. John Wiley & Sons, Inc., New York.
[21]Ghirardato, P. (1997). On independence for non-additive measures, with a Fubini theorem. J. Econom. Theory 73 261-291.
[22]HUBER, P. J. (1973). The use of Choquet capacities in statistics. Bull. Inst. Internat. Statist.45 181-191.
[23]HUBER, P. J. and RONCHETTI, E. M. (2009). Robust Statistics,2nd ed. Wiley, New Jersey.
[24]KEYNES, J. M. (1921). A treatise on probability. Macmillan, London.
[25]Le CAM, L. (1986). The central limit theorem around 1935. Statist. Sci.1 78-96.
[26]Li, W. and Chen Z.,(2011). Law of large numbers of negatively correlated random variables for capacity, Acta Mathematicae Applicatae Sinica, En-glish Series,27(4),749-760.
[27]Maccheroni, F., and Marinacci, M., (2005). The strong law of large numbers for capacities, The Annals of Probability,33 (3),1171-1178.
[28]Marinacci, M.,(1999) Limit laws for non-additive probabilities and their fre-quentist interpretation, Journal of Economic Theory,84,145-195.
[29]Peng, S.,(2007). Law of large number and central limit theorem under non-linear expectations, arXiv:0702358vl [math.PR].
[30]Peng, S., (2008). A new central limit theorem under sublinear expectations, arXiv:0803.2656vl [math.PR].
[31]Peng, S., (2009). Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expec-tations, Science in China Series A-Mathematics,52(7),1391-1411.
[32]PENG, S. (2010). Nonlinear expectations and stochastic calculus under un-certainy. arXiv preprint arXiv:1002.4546.
[33]Schmeidler, D., (1989). Subjective probability and expected utility without additivity, Econometrica,57,571-587.
[34]Teran, P. (2013). Laws of large numbers without additivity. Trans. Amer. Math. Soc. Accepted.
[35]Shapley, L., (1971). Cores of convex games, International Journal of Game Theory,1,11-26.
[36]Walley, P., and Fine, T. L. (1982). Towards a frequentist theory of upper and lower probability. Ann. Statist.10 741-761.
[37]Wasserman, L. A. and Kadane, J., (2009). Bayes'theorem for Choquet ca-pacites. Annal of Statistics,18,1328-1339.