Research On Limit Theorems Under Non-linear Expectation

In recent years,in order to solve some problems in statistics,risk measures,mathematical economics and other fields which are difficult to deal with by classi-cal probability theory,various non-linear probabilities and non-linear expectations appear and get a wide range of researches and developments.In order to solve the problems in statistical mechanics and potential theory,Choquet(1954)put forward the concepts of capacity and Choquet expectation.After that Choquet expectation was widely used in statistics and asset pricing in incomplete market.Peng(1997)proposed a non-linear expectation-g-expectation based on backward stochastic differential equation(BSDE).Later studies showed that g-expectation is suitable to describe the non-lineax risks in finance.Delbaen(1998),Artzner et al.(1999)introduced a new way of risk measure,a non-linear risk measure called coherent risk measures.To describe the volatility uncertainty in finance,Peng(2007a)initiated the concept of G-normal distribution with variance uncertainty based on the general sub-linear expectation,and then introduced a typical sub-linear expectation-G-expectation.By these studies we can find that the nature of the above non-linear expecta-tions is sub-linear expectation when they are applied to solve related problems in finance such as coherent risk measures and asset pricing in the incomplete markets.Therefore we can convert them into a more general sub-linear expectation when we study them.In addition,the classical laws of large numbers and central limit theorems play an important role in probability theory and mathematical statistics.As the appearance of capacities and non-linear expectations,the limit theory for capacities and non-linear expectations has always been the hot topic concerned and researched by scholars and widely used in non-linear risk measures,asset pricing and other fields.The classical proofs of limit theorems depend on the additivity of probability measure and expectation.Because the sub-linear expectation is no longer additive,most classic methods of proofs are not applicable which makes it difficult for us to prove limit theorems under sub-linear expectation.How to improve the previous results,obtain more accurate results about the laws of large numbers and central limit theorems under sub-linear expectation,then improve the sub-linear expecta-tion theory system,make it better to solve various problems in finance,economics,statistics,is worth thinking and of great significance.Motivated by the above problems,this doctoral dissertation mainly researches the laws of large numbers and central limit theorems under sub-linear expectation.The innovations of this article are as follows:1.We obtain two types of weak laws of large numbers for sub-linear expectation and get the equivalence relation between them.We find out that the mean value of random variables does not converge in probability to a certain value but weakly converge to the interval between lower and upper expectations with lower capacity 1 and it weakly converge to all the point in this interval with upper capacity 1.2.We study the relation between the sub-linear expectation and Choquet ex-pectation and find out the difference from the classical strong laws of large numbers that 1st moment condition for general sub-linear expectation can not ensure the validity of strong laws of large numbers.Then we obtain a strong laws of large numbers for sub-lineax expectation under controlled 1st moment condition.3.We obtain a strong law of large numbers for sub-linear expectation under a general moment condition and get two more precise results under this general moment condition by using the results of weak laws of large numbers.We illustrate that this general moment condition is the weakest moment condition to ensure the validity of strong laws of large numbers for sub-linear expectation.4.We obtain a series of strong laws of large numbers for sub-linear expectation without independence assumptions with a more general norming sequence {an}.5.We obtain the Berry-Essen bound for sub-lineax expectation and two dif-ferent forms of central limit theorems for sub-linear expectation with two different normalized sequences.This article is divided into six parts.The main results are as follows:(1)In Chapter 1,we introduce the research background and give the concepts and elementary properties of capacity,general sub-linear ex-pectation and some specific sub-linear expectations.Definition 0.1.1.A set function V:F?[0,1]is called a capacity,if it satisfies(1)v(?)=0,V(?)=1;(2)V(A)?V(B),A(?)B,A,B(?)F.Let(?,F)be a given measurable space.Let H be a subset of all random variables on(?,F)satisfying(1)H is a vector lattice,i.e.H is a linear space,all the constant(?)Definition 0.1.2.A sub-linear expectation E on H is a functional:(?)satisfying the following properties:for any random variables X,Y ?H,(1)Monotonicity:If X ?Y,then E[X]? E[Y];(2)Constant preserving:E[c]? c,(?)c?R;(3)Positive homogeneity:(?)(4)Sub-additivity:(?)The triple(?,H,E)is called a sub-linear expectation.We define the conjugateexpectation ? of E by:(?).We define the capacity V induced by(?)and the conjugate capacity(?)(?)(?)In Chapter 2,we investigate the weak laws of large numbers under sub-linear expectation.We first give the definition of Peng's independence,then for a sequence of independent random variables we obtain two forms of weak laws of large numbers under 1st moment condition and then get the equivalence relation between them:Definition 0.2.1.(Peng's independence)Let X=(X1,...,Xm)Xi?H and(?)be two random variables on(?,H,E).Y is said to be independent of X,if for each test function(?)we have(?)Theorem 0.2.2.Given a sub-linear expectation space(?)sequence of independent random variables satisfying(?)for any(?)Assume(?)n)]=0.Then we have(1)for any(2)for anyTheorem 0.2.3.Given a sub-linear expectation space(?,H,E).Let(?)be a sequence of independent random variables satisfying(?),for any n=1,2,....Let(?)Then the following two forms of weak laws of large numbers are equivalent:(1)for any test function ??Cb(R),(2)for anyand for anyTheorem 0.2.4.Given a sub-linear expectation space(?)be a sequence of independent random variables satisfying(?)for any(?)Assume(?)n)]=0.Then for any test function ??Cb(R),Then we generalize the above weak laws of large numbers to the situation that the 1st moment for sub-linear expectation is not exist or the upper expectations and lower expectations are not equal.Theorem 0.2.5.Given a sub-linear expectation space(?)be a sequence of independent random variables satisfying(?)(?)for any(?)Assume(?)Then we have(1)for any ?>0,(2)for any(3)if we further assumethen for any test functionTheorem 0.2.6.Given a sub-linear expectation space(?)be a sequence of independent random variables satisfying(?)(?)Let(?)Then the following two forms of weak laws of large numbers are equivalent:(1)for any test function(2)for any ?>0,and for any ?>0,any(?)satisfying(?)Theorem 0.2.7.Given a sub-linear expectation space(?,H,E).be a sequence of independent random variables satisfying for any Assume n)]=0.Then we have(1)for any ?>0,(2)for any ?>0 and(3)if we further assumethen for any test function ??Cb(R),Theorem 0.2.8.Given a sub-linear expectation space(?)be a sequence of independent random variables satisfying(?)for any n=1,2,...and(?)(?)Then the following two forms of weak laws of large numbers are equivalent:(1)for any test function ??Cb(R),(2)for any ?>0,and for any ?>0,any(?)satisfyingIt can be found that in our results the random variables do not have to be identically distributed.And these results will contribute to the proofs in Chapter 4.(?)In Chapter 3,we study the strong laws of large numbers for sub-linear expectation under 1st moment condition.We first give the definition of polar set,then analyze the relation between sub-linear expectation and Choquet expectation.Definition 0.3.1.Given a capacity V,a set A is said to be a polar set if V(A)=0.We call a property holds "quasi-surely"(q.s.)if it holds outside a polar set.Theorem 0.3.2.Given a sub-linear expectation space(?,H,E),V is the capacity induced by E and Cv is the Choquet expectation generated by V.Then we have(1)(?)(2)(?)(3)(?)(4)(?)The advantage of the above theorem is that we do not need to add any continu-ous assumption to sub-linear expectation.And this result reveals that the condition Cv[|X|]<? itself implies that the sub-linear expectation has certain degree of continuity:(?)Then by the above theorem,combining the result of Zhang(2016a)-the strong law of large numbers under 1st moment condition for Choquet expectation,we illustrate the 1st moment condition for sub-linear expectation can not ensure the validity of the strong law of large numbers.Theorem 0.3.3.Given a sub-lineax expectation space(?,H,E),the capacity V induced by E is lower semi-continuous.Let(?)be a sequence of independent identically distributed random variables with(?)(1)if(?)then(?)(?)(2)if we also assume V is upper semi-continuous,then(?)(?)implies Cv[|X1|}<?.By virtue of this theorem,we illustrate that we can always find some specific sub-linear expectation such that there exists a sequence of independent identically distributed random variables(?)satisfying E[|X1|]<? but Cv[|X1|]??,so the strong law of large numbers is not valid.At last we give a strong laws of large numbers under controlled 1st moment condition.Theorem 0.3.4.Given a sub-linear expectation space(?,H,E),the capacity V induced by E is lower semi-continuous.Let(?)be a sequence of independent random variables with(?)for any n?N*.Assume that there exists a random variable X?H satisfying(?)for any n?N*and(?)(?)(?)also(?)(?)(?)In Chapter 4,we discuss the weakest moment condition for strong laws of large numbers under sub-linear expectation.We first give a strong law of large numbers for sub-lineax expectation under a general moment condition.We declare that this class of moment conditions can maintain the validity of strong laws of large numbers and then obtain two more pre-cise results under this general moment condition by the weak laws of laxge numbers in Chapter 2.In addition,by the similar method in Chapter 3,we illustrate this general moment condition is the weakest moment condition to maintain the validity of strong laws of large numbers under sub-linear expectation.Definition 0.4.1.Let?c(?d)be a set of functions ?(x)defined on[0,?)in which?(x)satisfies:(1)?(x)is nonnegative and nondecreasing on[0,?)and positive on[x0,?)for some x0?0.Series(?)is convergent(divergent).(2)for any fixed ?>0,there exists a constant C>0 such that(?)for any x?x0.Theorem 0.4.2.Given a sub-linear expectation space(?,H,E),E is lower semi-continuous and V is the induced capacity.Let(?)be a sequence of indepen-dent random variables satisfying(?)for some(?)and(?)(?)(?)also(?)(?)Theorem 0.4.3.Given a sub-linear expectation space(?,H,E),E is lower semi-continuous and the induced capacity V is continuous.Let(?)be a sequence of independent random variables satisfying(?)for some(1)(2)We denote C({xn})as the cluster set of a sequence of {xn} on R.Then for any(?)Furthermore,by our main results we obtain the strong laws of large numbers of function extended form and generalize the strong laws of large numbers to the situation that the upper expectations and lower expectations are not equal.Corollary 0.4.4.Under the assumptions of theorem 0.4.2,for any continuous function ?(·)on R,we have(1)alsoIf we further assume that V is continuous,then(2)(3)for anyTheorem 0.4.5.Given a sub-linear expectation space(?,H,E),IE is lower semi-continuous and V is the induced capacity.Let(?)be a sequence of indepen-dent random variables satisfying(?)for some(?)and(?)(?)and thenIf we further assume that V is continuous,then(2)(3)for anyAs a corollary,we also obtain the weighted strong laws of large numbers.We denote ?w as a set of functions satisfying the conditions(1),(2)of ?c and the following condition(3):(3)for any fixed a>0,there exists C>0 such that(?)for any X?X0.Theorem 0.4.6.Given a sub-linear expectation space(?,H,E),E is lower semi-continuous and V is the induced capacity.Let(?)be a sequence of indepen-dent random variables satisfying(?)for some(?)and(?)be a sequence of uniformly bounded positive real numbers and set(?)Then(1)If we further assume that V is continuous,then(2)(3)for any(?)(?)In Chapter 5,we investigate the strong laws of large numbers under sub-linear expectation without independence assumptions.We give two strong laws of large numbers without independence assumptions with a more general norming sequence an and indicate that they both can be reduced to the i.i.d situation.Theorem 0.5.1.Given a sub-linear expectation space(?,H,E),the capacity V induced by E is lower semi-continuous.Let(?)be a sequence of nonnegative random variables and(?)be a sequence of nondecreasing unbounded positive numbers.Assume for all large sufficiently n,m,for all large sufficiently n,m.And for any p>0,for some function ???c,for some function ???c,Then alsoTheorem 0.5.2.Given a sub-linear expectation space(Q,H,E),the capacity V induced by IE is lower semi-continuous.Let(?)be a sequence of nonnegative random variables and(?)be a sequence of nondecreasing unbounded positive numbers.Assume E[Sn]=O(an)and for any p>0,Then alsoCorollary 0.5.3.Given a sub-linear expectation space(?,H,E),the capacity V induced by lE is lower semi-continuous.Let(?)be a sequence of nonnegative random variables satisfying(?)Then alsoIn addition,we obtain a strong law of large numbers without independence assumptions under conditions of quasi-sturely.Theorem 0.5.4.Given a sub-linear expectation space(?,H,E),the capacity V induced by E is lower semi-continuous.Let(?)be a sequence of random variables.Assume that for any n?1 and some 0??<1,there exists some C>0 such thatThen also(?)In Chapter 6,we study the central limit theorems under sub-linear expectation without identical distribution assumptions.We first give the definition of G-normal distribution and obtain two central limit theorems with respect to two kinds of normalized sequences Sn/Bn and Sn/Bn under the conditions similar to Lyapunov's and indicate that they both can be reduced to the i.i.d situation.Definition 0.6.1.(G-normal distribution)Let X be a 1-dimension random variable on a given sub-linear expectation space(?,H,E)with(?)X is said to be G-normal distributed,if there holds for each X which is a independent copy of X.We denote it by(?)Theorem 0.6.2.Given a sub-linear expectation space(?)sequence of independent random variables.Assume:(1)(2)? is G-normal distributed,satisfying Let Then for any h>0 and anyfixed(?)there exists some 0<?<1,some constant C>0 and some constant Ch>0 depending on h such thatTheorem 0.6.3.Given a sub-linear expectation space(?,H,E),(?)is a sequence of independent random variables.Assume:(1)(2)? is G-normal distributed,satisfying(3)? satisfies(4)for any 0<?<1,Then for any(?)we haveTheorem 0.6.4.Given a sub-linear expectation space(?)is a sequence of independent random variables.Assume(1)(2)? is G-normal distributed,satisfying(3)? satisfies(4)for any 0<?<1,(?)Then for any(?)we have(?)(?)The advantage of our results is that(2+?)th moments and 2nd moments of random variables do not have to be finite uniformly.