| Since Bernoulli described the stability presented in large scale observations and proposed the famous limit theorem known as’Bernoulli theorem’in 1713, the study of probability has been carried out for more than 300 years. In late 1800s, limit theory became the pivotal topic in the probability field by the generalization of limit theorems made by Chebyshev, Markov and other Russian mathematicians. In 1933, Kolmogorov initialed six axioms of probability through the approach of measure theory, which lays the foundation for modern probability theory. After the axiomatization, the probability theory has been widely applied in various fields and witnessed a rapid development.The classic laws of large number and central limit theorems are based on the linearity of expectations and probability measures. However, as the development of economy and finance, many phenomena can not be well explained by those the-ories due to the traditional linear set-up. Take financial derivatives for instance, huge profits come with comparative large risks. Considering the fact that most risk behavior of derivatives exhibit non-linearity, the traditional linear approaches are not sufficient for evaluating, managing and controlling those risks. The limitation of linear methods may cause poor estimation of risk and sequentially lead to unde-sirable results ranging from economic losses of banks and financial institutions to national and global financial crisis. Motivated by the deficiency of linear measures and expectations, numerous scholars have explored various non-linear theories to describe and measure those risks more accurately for recent decades.Delbaen[41], Artzner and Delbaen[3] initials a kind of non-linear risk measures called coherent risk measure which draws others’attention to the topic of non-linear measures. Pardoux and Peng[70] provide properties of the solution of backwards stochastic differential equations (BSDE): yt=ζ+∫tTg(ys,zs,s)ds-∫tTzsdWs and Peng[73] propose the g-expectation and g-conditional expectation on the basis of BSDE in the sense that properties of g-expectations will be determined by the generators g of BSDE. Gianin[53] builds a relation between risk measures and g-expectations while providing the properties of static and dynamic risk measures induced by g-expectations and g-conditional expectations. Also, Gianin[53] gives reasonable financial explanations for risk measures based on g-expectations. Coquet et.al.[9] and Chen et.al.[30] studies the properties of a special kind of g-expectations generated by generators with the form g=μ|z|. Chen and Epstein[31] establish the relation between multiple prior and g-expectation and obtain so called Chen-Epstein formula for asset pricing. Their results develop the rational expectation pricing theory proposed by the Nobel winner Lucas and explain the Allais paradox and stock premium mystery by proving that the asset price is the sum of system price and price of ambiguity. Above results indicate that nonlinear g-expectation theory is an effective tool to describe and measure risk behaviors.More generally, Peng[75] defines the sublinear expectations directly as real val-ued functions satisfying monotony, constant preserving, sublinearity and positive homogeneity without relying on according probability measures. Combined with PDE theories, Peng establishes a complete sublinear theoretic system by defining the maximum distribution, G-normal distribution, G-Brownian motion and G-BSDE and [44,75,78,80]etc. prove the law of large numbers, central limit theorems and the Ito formula. Gong et.al.[107,108] applies the sub-linear expectation theory in the study of VAR (Value at Risk) and obtains two indexes for risk measurement called R-VAR and R-ES, which provides innovative theoretical and solid practical supports for prudential risk management with uncertainty.Under traditional linear probability theory, there is a one-to-one correspon-dence between the probability measure and the expectation. Therefore, intuitive-ly, another optional tool for describe those non-linear risks will be the non-linear probability theory. However, under the non-linear framework, the one-to-one corre-spondence between probability measures and expectations no longer exists. Given a certain non-linear expectation, we can uniquely derive a non-linear probability but the opposite does not necessarily hold. Thus the non-linear expectation theory and non-linear probability theory are two related techniques which are proceeded in different ways due to the lack of correspondence.Choquet[11] brought up the definition of capacity in 1954, which can be seen as a special case of non-linear probability measures. Choquet expectations are obtained based on capacities. Numerous scholars extend linear probability theories to the capacity cases (see [35,67,94] etc.). Chen et.al.[28,29,36] obtains important properties of a special kind of capacities, called the upper probability, generated by taking maximal values among a family of linear probability measures and derives a law of large numbers under it. Compared with the law of large numbers under linear framework, for the upper probability, the sample mean of given random variables will fall into an interval instead of converging to one single point.Motivated by above works, the main purpose of this paper is to investigate and generalize various limit theorems under nonlinear framework to supplement the nonlinear theoretic system theoretically and practically.This paper can be divided into six parts:(I) In Chapter 1, we introduce several types of nonlinear probabilities and expectations in Section 1. And in Section 2, we investigate properties of quasi-surely convergence and convergence in capacity of random variables under upper-probability. In addition, we achieve Kolmogorov inequality, Rademacher inequality and related results.In Chapter 1, we introduce upper probability, sublinear expectation, Choquet expectation,g-expectation and the relationship of different nonlinear probabilities and expectations in Section 1. For details, see the text. In Section 2, we investigate quasi-surely convergence and convergence in capacity under nonlinear probabilities, and extend several theorems from classic probability to upper probability. We only list the main results here. Suppose (Ω,F, P, E) to be an upper-expectation space.Theorem 0.1.1. If X_n converge in capacity to X, then there exists a sequence of positive integers nk'∞ oo such that X_nk'X q.s..Lemma 0.1.2. Let{X_n}n=1∞ be a sequence of random variables under upper-expectation space (Ω, F, P, E). If there exits random variables X and a sequence of positive integers nk↑∞ such that X_nk'X q.s. and then X_n'X q.s.Theorem 0.1.3. (Kolmogorov inequality)Let{X_i}i=1∞ be a sequence of in-dependent random variables under upper-expectation space(Ω,F,P,E)such that E[X_i]=ε[X_i]=0,E[X_i2]<∞ for any i≥1.Denote Sk=∑i=1k X_i.Then,for any ε>0,we have In addition,if there exits constant C such that |X_i|≤C for any 1≤i≤n,thenTheorem 0.1.4.Let{X_i}i=1∞ be a sequence of independent random variables under upper-expectation space(Ω,F,P,E)such that E[X_i]=ε[X_i]=0 for any i≥1 and satisfying ∑i=1∞ E[X_i2]<∞,Then S_n=∑i=1n X_i converge quasi-surely.Theorem 0.1.5. Let{X_i}i=1∞ be a sequence of random variables under upper-expectation space(Ω,F,P,E)such that E[X_iXj]≤0 for any i≠j.Let{bn}n=1∞ be a a positive sequence increasing monotonely to infinity and such that ∑n=1∞ bnE[X_n2]< ∞.For any k≥1,set nk to be the smallest integer n such that bn≥k.Then S_nk converge quasi-surely.Theorem 0.1.6.(Rademacher inequality)Let{X_i}i=1∞ be a sequence of random variables under upper-expectation space(Ω,F,P,E)such that E[X_iXj]≤0 for any i≠j.ThenTheorem 0.1.7. Let{X_i}i=1∞ be a sequence of random variables under upper-expectation space(Ω,F,,P,E)such that E[X_iXj]≤0 for any i≠j and ∑n=1∞(log n)2E[X_n2]<∞.Then S_nk converge quasi-surely.Theorem 0.1.8.Let f and g be two functions defined on Ha,n.If all the following assumptions are satisfied,then S_n converge quasi-surely.(Ⅱ) In Chapter 2, we define the concept of asymptotically almost neg-atively associated random variables under upper-expectation space. In this framework, we prove two types of Rosenthal’s inequality with the subadditive property of expectation instead of additivity. Finally, we prove a strong law of large numbers as the application of our Rosenthal’s inequality. Suppose that (Ω,F, P,E) be an upper-expectation space and V be the upper-probability.Definition 0.2.1. (Asymptotically almost negatively associated) A sequence {X_n}n=1∞ of random variables is called asymptotically almost negatively associat-ed (AANA) under E if there exists a nonnegative sequence {η(n)}n=1∞ such that limn'∞ η(n)= 0 and for all n, k≥1 and for all coordinatewise nodecreasing or nonincreasing continuous functionsf and g whenever the expectations exist. And {η(n)}n=1∞ are called mixing coefficients.Lemma 0.2.2. Let 1/p+1/q=1, p> 1, q> 1 and{X_n}n=1∞ be an AANA sequence of random variables with mixing coefficients{η(n)}n=1∞, then for all n, k≥1 and coordinatewise nondecreasing or coordinatewise nonincreasing functions.Theorem 0.2.3. (Rosenthal inequality(a))Let 1/p+1/q=1,1<p≤2 and {X_n}n=1∞ be an AANA sequence of random variables under E with E[X_n]=ε[X_n]= 0. And {η(n))n=1∞ are the corresponding mixing coefficients. Then there exists a positive constant Cp depending only on p such that for any n≥1, In particular,if ∑n=1∞ η2(n)<∞,then for any n≥1,Theorem 0.2.4.(Rosenthal inequality(b))Let1/p+1/q=1,p≥2 and {X_n)n=1∞ be an AANA sequence of random variables under E with E[X_n]=ε[X_n]=0. And {η(n)}n=1∞ are the corresponding mixing coefficients.Then there exist positive constants Cp and Cp’ depending only on p such that for any n≥1, In particular,if ∑n=1∞ ηq/p(n)<∞,then for any n≥1,As application of our Rosenthal inequality,we achieve a SLLN at the end of this chapter.Theorem 0.2.5.Let1/p+1/q=1,1<p≤2,b1,b2,…be a nondecreasing unbound-ed sequence of positive numbers and {X_n}n=1∞ be an AANA sequence of random variables under E with E[X_n]=ε[X_n]=0.And {η(n))n=1∞ are the corresponding mixing coefficients.If ∑n=1∞ η2(n)<∞ and ∑n=1∞E[|X_n|p]/bnp <∞,then limn'∞S_n/bn=0q.s. (Ⅱ) In chapter 3, under upper-expectation space, we investigate two types of strong laws for weighted sums of vertical independent random variables. We also discuss the stability of random variables and derive a Strassen type invariance principle as applications of our results. At last, we give a Marcinkiewicz-Zygmund type strong law of large numbers for negatively associated random variables.Suppose that (Ω, F, P,E) be a upper-expectation space. Definition 0.3.1. (Vertical independence) Let X1,X2,…, X_n+1 be real mea-surable random variables on upper-expectation space (Ω,F, P,E). X_n+1 is vertical independent of (X1,…,X_n), if for each nonnegative measurable function φi(·) on R with E[φ1(X_i)]<∝, i=1,…, n+1, we have {X_n}n=1∞ is said to be a sequence of vertical independent random variables, if X_n+1 is independent of (X1, X2,…, X_n) for each n∈N*. Theorem 0.3.2. (Strong laws for weighted sum (a)) Let{X_i}i=1∞ be a se-quence of vertical independent random variables under upper-expectation space (Ω,F,P,E). supi≥1E[|X_i|α+1]<∞ for some constant a> 0. Let{ai}i=1∞ be a sequence of bounded positive numbers and set An=∑i=1n ai.Suppose there exists a constant β∈(0, min(1,α)) such that Then, alsoTheorem 0.3.3. (Strong laws for weighted sum (b)) Let{X_i}i=1∞ be a se-quence of vertical independent random variables under upper-expectation space (Ω, F, P,E). Suppose that supi≥1 E[|X_i|α+1]<∞ for some constant α>0. Let {ai}i=1∞ be a sequence of bounded positive numbers and set An=∑i=1n ai.Suppose there exists a constant β∈(0,min(1,α)) such that Then, Theorem 0.3.4.(Invariance principle) If all the assumptions in Theorem 0.3.2 are satisfied, then for all continuous functions φ(·) on R, we have Definition 0.3.5.(Negatively associated) Let {X_n}n=1∞ be a sequence of random variables on upper-expectation space (Ω,F, P, E). Suppose that A and B are two nonempty and disjoined subsets of{1,2,…, n} satisfying i< j for any i∈A and j∈B.{X_n}n=1∞ is called a sequence of negatively associated (NA) random variables ifE[f(X_i:i ∈A)g(Xj:j∈B)}≤E[f(X_i:i∈A)]·E[g(Xj:j∈B)] for all nonnegative coordinatewise nodecreasing or nonincreasing continuous func-tions f and g.Theorem 0.3.6. (Marcinkiewicz-Zygmund SLLN) Let{X_i}i=1∞ be a sequence of NA random variables on upper-expectation space (Ω,F, P, E), satisfying supi≥1 E[|X_i|α+1]< ∞ for some constant 0< a< 1. Then for any 1≤p<1+α, we have (IV) In Chapter 4, we discuss the concept and proporties of convolu-tionary random variables under upper-expectation space. Then we derive several limit theorems for convolutionary random variables under Fatou type capacity. Also, we obtain the Borel-Cantelli lemma for convolution-ary random variables.Let (Ω,F, P, E) be an upper-expectation space. Definition 0.4.1. (Convolutionary) Random variables X and Y are said to be of convolution under E if for any φ∈Cb(R) Definition 0.4.2. Let V be a capacity defined on F to [0,1]. V is called to be a Fatou type capacity if for any An ∈F, we haveAll the limit theorems in Chapter 3 can be easily extend to convolutionary random variables under Fatou type capacity. We omit them here. Furthermore, the following renewal theorem can be derived.Theorem 0.4.3. Let{X_i}i=1∞ be a sequence of nonnegative convolutionary random variables under upper-expectation space (Ω,F, P,E) and supi≥1,E[|X_i|α+1]<∞ for some constant a>0. Suppose that E[X_i]=μ,ε[X_i]=μ for any i=1,2,…, and 0<μ<μ. Set S_n:=∨i=1n=1X_i,S0=0. Define Then,Theorem 0.4.4. (Borel-Cantelli lemma) Let V be a Fatou type capacity and {Ai}i=1∞ be a sequence of events in F. If{IAi}i=1∞ be a sequence of convolutionary random variables under E and ∑i=1∞V(Ai)=∞,then (V)In Chapter 5,inspired by Peng’s centrallimit theorem under sublin-ear expectation,we investigate several generalized central limit theorems under weaker conditions with the notion of G-normal distribution. We first loose the mean condition E[X_n]=ε[X_n]=0 to |E[X_n]|+|ε[X_n]|=O(1/n). And then weaken some moment conditions by truncation of random vari-ables.Suppose(α,H,E)be a sublinear expectation space.We use the definition of G-normal distribution derived by Peng to obtain our central limit theorems. Definition 0.5.1.(G-normal distribution)A random variable X in the sublinear expectation space(Ω,H,E)with E[X2]=σ2 and ε[X2]=σ2 is said to be G-normal distributed,denoted by X～N(0;[σ2,σ2]),if for each Y which is a independent copy of X,it holds that aX+by～(?)X,(?)a,b≥0. Theorem 0.5.2. (CLT(a))Let{X_n}n=1∞ be a sequence of independent random variables under sublinear expectation space(Ω,H,E)satisfying the following con-ditions: Then the sequence {S_n/(?)n=1∞,where S_n=∑i=1n X_i,converges in law to the G-normal distribution,that isTheorem 0.5.3.(CLT(b))Let {X_n}n=1∞ be a sequence of independent sequence under sublinear expectation space(Q,H,E)satisfying the following assumptions:Then the sequence{S_n/(?)n=1∞,where S_n=∑i=1nX_i,converges in law to the G- normal distribution, that is(VI) In Chapter 6, we introduce the concepts of upper-lower set-valued probabilities and related upper-lower set-valued expectations for random variables. With a new concept of independence for random variables, we show a strong law of large numbers for upper-lower set-valued probabili-ties. Furthermore, we extend those concepts and theorem to the case of fuzzy-set.Definition 0.6.1. (Upper-lower set-valued probabilities) Let{Πn}n=1∞ be a se-quence of nonatomic closed set-valued probabilities. Define two set-valued functions from F to P([0,1]) by Γ(·)=∪n=1∞ Πn(·) and Γ(·)=∩n=1∞ Πn(·) respectively. (Γ,Γ) is called a pair of upper-lower set-valued probabilities if for any A ∈F,∩n=1∞ Πn(A) is a convex set and ∩n=1∞ Πn(A) is a nonempty convex set. Definition 0.6.2. (Upper-lower set-valued expectations) Let (Ω, F, Γ) be an upper set-valued probability space and Γ(·) be the lower set-valued probability corresponding to Γ(·). Let X:Ω'R be a real random variable. Then the upper expectation Er[X] and lower expectation EΓ[X) are defined respectively byDefinition 0.6.3. (Independence) Let X1,X2,…,X_n+1 be real measurable ran-dom variables on (Ω,F). X_n+1 is said to be independent of (X1, X2,…, X_n) under Er, if for each nonnegative measurable function φi(·) on R such that{EΓ[φi(X_i)]}i=1n+1 are bounded sets, we have where the product of sets A and B is defined by AB={ab:a∈A,∈GB}.{X_n}n=1∞ is said to be a sequence of independent random variables under EΓ if X_n+1 is independent of(X1,X2,…,X_n)for each n≥1. Theorem 0.6.4.(SLLN under upper set-valued probability)Let{X_i}i=1∞ be a sequence of independent random variables under EΓ. Suppose for some α>0, {EΓ[|X_i|1+α])i=1∞ are all bounded sets and EΓ[X_i+]=EΓ[X1+],EΓ[X_i-]=EΓ[X1-], EΓ[X_i+]=EΓ[X1+],EΓ[X_i-]=EΓ[X1-]for any i≥1.Then almost surely with respect to Γ.Definition 0.6.5.(Upper-lower fuzzy-set valued probabilities)Let{μn)n=1∞ be a sequence of f-probabilities.u(·)=∪n=1∞ μn(·)and u(·)=∩n=1∞ μn(·)are two set-valued functions from F to FC([0,1]).(u,u)is called a pair of upper-lower f-probabilities if for any A∈F,α∈(0,1],∪n=1∞ μnα(A)is a convex set and ∩n=1∞ μnα(A) is a nonempty convex set.Definition 0.6.6. (Upper-lower fuzzy-set valued expectations)Let u(·)= ∪n=1∞ μn(·)and u(·)=∩n=1∞ μn(·)be upper f-probability and lower f-probability respectively,where{μn}n=1∞ is a sequence of f-probabilities.If for any α∈(0,1], Lα(∪n=1∞ Eμn[X])is closed set,then the fuzzy sets (∪n=1∞ Eμn[X])(denoted by Eu[X]) and (∩n=1∞ Eμn[X])(denoted by Eu[X])can be defined as the upper and lower expec-tations of random variable X respectively,that is where{Eμn[X])n=1∞ are fuzzy sets such that Lα(Eμn[X])=Eμnα[X]for any α∈(0,1]. Definition 0.6.7. Let X1,X2,…,X_n+1 be real measurable random variables on (Ω,F).X_n+1 is said to be independent of(X1,X2,…,X_n)under Eu,if for each non-negative measurable function φi(·)on R with that{Esuppu[φi(X_i)])i=1n+1 are bounded sets,we have {X_n}n=1∞ is said to be a sequence of independent random variables under Eu,if X_n+1 is independent of(X1,X2,…,X_n)for each n≥1.Theorem 0.6.8. (LLN under upper fuzzy-set valued probabilities)Let {X_i}i=1∞ be a sequence of independent random variables under Eu.Suppose that for some β>0,{Esuppu[|X_i|1+β])i=1∞ are all bounded sets,and Eu[X_i+]=Eu[X1+], Eu[X_i-]=Eu[X1-],Eu[X_i+]=Eu[X1+],Eu[X_i-]=Eu[X1-]for any i≥1.Then... |
PDF Full Text Request