Laws Of Large Numbers For Non-linear Probabilities And Related Problems

Classical probability theory,as is well known,is based on measure theory,for which additivity and continuity are natural and crucial assumptions.Non-linear probabilities(capacities)or non-additive probabilities,on the other hand,are mono-tone set functions for which additivity is not satisfied.The first fundamental contri-bution in the field of non-additive set functions was made by Choquet[12]in 1953 in his research of potential theory and statistical mechanics,where he introduced the integral named later after him.Capacity theory was also studied by Shapley[53,54]in the same year in the perspective of game theory,and a result therein is that any convex games have non-empty cores,where the convex games are actual-ly supermodular capacities.This work is generally considered as one of the most important contributions in game theory and in economic theory.With the development of decision theory,modern economic theory and the fi-nance,it has been found that many uncertain phenomena cannot be well modeled under the framework of additive probabilities.In order to solve these problems,researchers have studied and made contributions in the field of non-additive prob-abilities,for example,Schmeidler[55,56],Gilboa[32,33],Denneberg[22],Ghi-rardato[34],Epstein[25,26].In recent years,Peng[45,49]initiated the notions of g-expectations and general sub-linear expectations,and made a series of contribu-tions;see further[46,48,50,51].Unlike the classical idea of defining integral via measure,Peng's concept of sub-linear expectation does not depend on any specific measure or capacity.In view of the fact that nonlinear expectation is not necessarily uniquely defined by the corresponding nonlinear probability(see[4],[13]),Peng's work provided a new approach to generalize and perfect the non-linear probabil-ity theory.Motivated by Peng's work,Chen and Epstein[6]established relations between g-expectations and problems in asset pricing,and applied the theory to explain the uncertainty and the Allais paradox therein.So far the theory of non-linear probabilities and expectations have attracted attentions of many researchers.Lots of results have been obtained in this field and have been applied widely in economics,statistics,even quantum mechanics,and especially as powerful mathe-matical supports in describing and measuring risk behaviors in finance.A systematic treatment of non-linear probability theory is getting more and more desirable,after its theoretical development as well as its applications over the years.Recall that limit theory lies in the center of classical probability theory,for which the law of large numbers(LLN in brief)plays a crucial role.In classi-cal sense LLN describes the phenomenon that the sample mean,obtained from a large number of trials,tends to the expected value.The first rigorous proof of LLN was obtained by Jacob Bernoulli in 1713,and this specific form of LLN is gener-ally known as Bernoulli's Theorem.After Bernoulli,many mathematicians made great contributions in this field,for example,Chebyshev,Markov,Borel,Cantelli,Kolmogorov and Khinchin.In the non-additive case,however,many limit laws are usually stated in different forms and are more complicated to prove,and therefore in many cases classical methods do not work.For instance,in the classical probability theory,if an infinite sequence {Xn}n?1 of identically distributed random variables with E[X1]<? satisfies(?)then we say that LLN holds for {Xn}n?1.Khinchin's Theorem states that LLN holds for a sequence of IID random variables with E[X1]<?.Now let v be a capacity defined on a measurable space(?,F).If(?),(0.4)then we say that LLN(under capacity v)holds for {Xn}n?1.Here??Cv[X1]?(-Cv[-X1]),?=Cv[X1]?(-Cv[-X1]),where Cv denotes the Choquet integral with respect to v.The convergence interval may become smaller,if the capacity satisfies some special properties.In particular,if v is a probability measure,then Cv[X1]=-Cv[-X1]=Ev[X1],and(0.4)reduces to(0.3).In resent years,many authors have worked on LLNs under non-linear assump-tions and obtained various results.For example,Marinacci[41],Maccheroni and Marinacci[40]obtained LLNs for totally monotone capacities,and Peng[49],Chen[5],Hu and Chen[9]derived various forms of LLNs on the sub-linear expectation space.In addition,Chen,Wu and Li[11]showed a strong LLN for upper and lower probabilities.Inspired by the earlier works,in this thesis we will investigate the laws of large numbers and related problems under the non-linear framework,and make further progresses and improvements.Our results can be summarized as follows:(?)We will work on LLN for submodular/supermodular capacities.Compared with the weak law of Marinacci[41],the space in our case may be non-compact,and the random variables may be discontinuous or irregular.(?)We will investigate strong LLN for submodular/supermodular capacities.This result generalizes the strong LLN of Maccheroni and Marinacci[40]whose results are confined to totally monotone capacities on a polish basic set.Our result is more general,since total monotonicity of capacities is a stronger version of supermodular.In addition,we do not require the basic space to be a polish space.(?)We will establish a general form of LLN for the upper-lower probabilities.The upper probabilities,sometimes also called upper envelopes of a class of probability measures,are a kind of typical sub-additive capacities.Compared with earlier results,our result holds under a rather weak assumption of random variables.(iv)We will eatablish two more forms of LLN for upper-lower probabilities,one of which is in the form of Peng's LLN.Compared with(iii),this work requires an additional technical assumption of time-consistence.(v)We will show LLN for sub-linear expectations and give applications in risk measures.Unlike the earlier results,our result is proved under a weak as-sumption instead of the finite moment assumption.This thesis consists of three chapters.In Chapter I,we will introduce capacities and Choquet integrals in Sections 1-3,and then in Section 4 give several lemmas to prove a convolution inequality for independent random variables.In Sections 5 and 6,we will prove a weak law of large numbers and establish a strong law of large numbers for submodular capacities,respectively.To state these theorems in detail,we need some definitions.Definition 0.1.1 Let ? denote the basic set,and S(?)2? be a set system with{(?),?)?S.A set function v on S is called a capacity,if it satisfies(1)v((?))=0,v(?)1;(2)(?)A,B ? S,A(?)B implies v(A)? v(B).If S is closed under complements and v is a capacity on S,then the conjugate capacity v on S is define by Definition 0.1.2 Let v be a capacity on S,and X:??R satisfy {???:X(?)>?}? S for any ??R.The Choquet integral of X with respect to v is defined by We say that X is integrable with respect to v,or the Choquet integral exists,if Cv[|X|]<?.Definition 0.1.3 Let(?,F)be a measurable space,and v a capacity on F.The random variables X,Y are called independent with respect to v,if for any sets A,B?B0(R),we have v(X?A,Y?B)=v(X?A)v(Y?B).Now we state the first result of Chapter ?.Theorem 1.1(LLN)Let(?,F)be a measurable space,and(v,v)a pair of con-jugated capacities on F,where v is submodular and continuous from below.Let{Xn}n?1 be a sequence of IID random variables with respect to v,and satisfy Cv[|X1]<?.Put Sn:= ?i=1n Xi.Then for any ?>0,we have(1)(?)(2)(?)(3)(?)The next theorem gives a generalization of Theorem 1.1 to the case that the random variables are not identically distributed.Theorem 1.2(LLN)Let(?,F)be a measurable space,and(v,v)a pair of con-jugated capacities on F,where v is submodular and continuous from below.Let{Xn}n?1 be a sequence of integrable and independent random variables with respect to v,and satisfies(?)The;n,for any ?>0,(1)(?)(2)(?)(3)(?)The following result gives a general form of strong LLN for v-q.s.bounded IID random variables.Theorem 1.3(SLLN)Under the assumption of Theorem 1.1,if {Xn}n?1 is v-q.s.bounded,then we have(1)(?)(2)(?)(3)(?)The next theorem gives a generalization of Theorem 1.3.Theorem 1.4(SLLN)Under the assumption of Theorem 1.2,if {Xn}n?1 is v-q.s.uniformly bounded,then we have(1)(?)(2)(?)(3)(?)In Chapter ? of this thesis we will introduce a special type of sub-additive capacities,called upper probabilities,and then give some new definitions.The main results of this chapter include a general form of LLN for the upper-lower probabilities without independence,and two more forms of LLNs under additional conditions.Definition 0.2.1 Let(?,F)be a measurable space,and P a class of probability measures on F.Define a set function on F as follows:V(A):=sup P(A),v(A):= inf P?P P(A),A?F.Then V and v are conjugated capacities on F.We call the triple(?,F,P)an upper-lower probability space.In addition,V and v are called the upper probability and the lower probability generated by P,respectively.Let(?,F,P)be given and L denote the class of all F-measurable random variables.Definition 0.2.2 The upper expectation and lower expectation generated by P are defined by E[X]:= sup P?P Ep[X],?[X]:= infP?P Ep[X],X ?L.respectively.Definition 0.2.3 Let {Xn}n?1 be a sequence of F-measurable random variables satisfies E[|Xi|]<?.We say that {Xn}n?1 is(recursively)weakly independent,if for any n>1,there holds E[Xn|Fn-1]? E[Xn],and?[Xn|Fn-1]??[Xn],where Fn =?(X1,X2,…,Xn).In this chapter,we first prove the following general form of LLN on the upper-lower probability space.Theorem 2.1(LLN)Let(?,F,P)be an upper-lower probability space.Suppose that {Xn}n?1 is a sequence of weakly independent random variables such that(a)E[Xn]=?,?[Xn]=?,(?)?1;(b)lim supn??E[|Xn]<?;(c)for any constant c>0,Set Sn:=?i=1n Xi.Then,for any ?>0,Note that the convergence interval in Theorem 2.1 is smaller than we consid-ered in Chapter ?,since for any random variable with Cv[|X|]<? there always holds where CV and Cv denote the Choquet integrals with respect to V and v,respectively.If we assume further that E is time-consistent with respect to {Xn}n?1,then we obtain the following stronger results.Theorem 2.2(LLN)Assume the conditions of Theorem 2.1,and let IE be time-consistent with respect to {Xn}n?1.Then we have the following:(1)for any x*?[?,?]and ?>0,we have(2)if there exist a,b ?R such that then[a,b](?)[?,?].Theorem 2.2 implies that[?[X1],E[X1]]is the smallest interval such that Sn/n converges under the lower probability v,and each point in[?[X1],E[X1]]is a limit-ing point of Sn/n under the upper probability V.The next result gives a special form of LLN.Theorem 2.3(LLN)Under the assumption of Theorem 2.2,for any function ??Cb(R),one has(?)In Chapter ?,we will introduce sub-linear expectations initiated by Peng.Our main result in this chapter is a general form of LLN under sub-linear expectations,without assuming finite moments.In addition,we give several applications of this LLN in risk measures.Definition 0.3.1 Let H be a linear space of random variables.A functional E:H?R on is a sub-linear expectation,if for any X,Y?H the following four conditions are satisfied:(1)(constant preserving)E[c]= c,(?)c?R;(2)(monotonicity)X?Y(?)E[X]?E[Y];(3)(positive homogeneity)E[AX]= ?E[X],A>0;(4)(sub-additivity)E[X+Y]?E[X]+E[Y].If E is a sub-linear expectation on H,then the triple(?,H,E)is called a sub-linear expectation space.Definition 0.3.2 Let(?,H,E)be a sub-linear expectation space.The conjugate expectation of IE is defined as?[X]:=-E[-X],X?H.The capacities induced by E and ? are defined respectively as V(A):=E[1A],v(A):=?[1A],A?F.Definition 0.3.3(Peng's independence)Let X =(X1,…,Xm)and Y =(Y1,…,Yn)be random vectors on(?,H,E).We say that Y is independent of X,if for any??Cl,Lip(Rm×Rn),E[?(X,Y)]=E[EE[?(x,Y)]|x=x].If for each n ? 1,Xn+1 is independent of(X1,…,Xn),then we call {Xn}n?1 an independent sequence of random variables.Definition 0.3.4(Peng's identical distribution)Let X =(X1,…,Xn)be a random vector on(?1,H1,E1),and Y =(Y1,…,Yn)a random vector on(?2,H2,E2).We say that X,Y is identically distributed,if for any ??Cl,Lip(Rn),E1[?(X)]=E2[?(Y)].Our main result in Chapter ? is the following theorem.Theorem 3.1(LLN)Let(?,H,E)be a sub-linear expectation space,and {Xn}n?1 a sequence of IID random variables satisfying(?)Then for any ?>0,we have(?).