Strong Laws of Large Numbers for Sublinear Expectation under Controlled 1st Moment Condition?

Cheng HU

Abstract This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition.For a sequence of independent random variables,the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite.By discussing the relation between sublinear expectation and Choquet expectation,for a sequence of i.i.d random variables,the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.

Keywords Sublinear expectation,Strong law of large numbers,Independence,Identical distribution,Choquet expectation

1 Introduction

In recent years,motivated by some problems in mathematical economics,statistics and financial mathematics,more and more researches on nonlinear expectation have appeared.People use nonlinear expectation to describe some phenomena in these fields which are difficult to be modeled exactly by classical probability theory.Choquet[5] first introduced the definition of capacity and it has been used in many fields of applied mathematics.To deal with the problems in risk measures,super-hedge pricing and modeling uncertainty in finance,Peng[12]initiated the definition of general sublinear expectation and the notion of independence and identical distribution for sublinear expectation.See more applications of nonlinear expectation for example,[3,7,11,13,15].

The major requirement for any probability theory is to give a frequentist justification to probability numbers via limit frequencies.The classical strong laws of large numbers(SLLNs for short)as fundamental limit theorems in probability theory play an important role in the development of probability and its applications.So the question arises naturally whether the SLLNs can be maintained in nonlinear expectation framework.There has been increasing interest in the investigation of SLLNs for nonlinear expectation.Marinacci[10]proved the SLLNs for a sequence of i.i.d random variables with respect to a totally monotone and continuous capacity under a multiplicative notion of independence.Maccheroni and Marinacci[9]introduced the definition of pairwise independence and proved a SLLN for a sequence of bounded i.i.d random variables with respect to a totally monotone capacity.They both indicate that any cluster point of empirical averages lies between the upper Choquet expectation CV[X1]and the lower Choquet expectation Cv[X1]with probability one under capacity v.That is

See more results regarding SLLNs for nonlinear expectations for example,[1–2,4,6,8].

Moreover,the gap between the Choquet expectations CV[X1]and Cv[X1]is bigger than that of the the upper expectation E[X1]and lower expectation E[X1].Chen et al.[4]obtained a more precise SLLN for a sequence of independent random variables under conditions of finite(1+α)-th moment for upper expectation.That is

Zhang[16]derived a SLLN of the above form for a sequence of negatively dependent identically distributed random variables under conditions of finite 1st moment for Choquet expectation.As we mentioned above,in sublinear situation,the Choquet expectation is larger than the upper expectation.And in classical probability theory,for a sequence of i.i.d random variables,the finiteness of 1st moment is the sufficient condition of the SLLNs.Our purpose in this paper is to study the SLLNs under some conditions of finite 1st moment for sublinear expectation.

First we obtain a SLLN under the condition that there is a random variable X such that for everyq.s.where X satisfiesThis assumption looks like a control condition which is weaker than the uniform boundedness,but is stronger thanFurthermore,we discuss whether the SLLN can be maintained under conditions that the uniform 1st moment for sublinear expectation is finite.But by discussing the relation between Choquet expectation and sublinear expectation and putting forward a counterexample,we find out that the SLLN may not be valid when only the uniform 1st moment for sublinear expectation is bounded which in turn reveals the fundamental difference between classical probability and sublinear expectation.

The rest of this paper is organized as follows.In Section 2 we recall some basic concepts of sublinear expectation and some useful lemmas.In Section 3 we give our main result,the SLLN for sublinear expectation under controlled 1st moment condition.In Section 4 we give a counterexample to illustrate that the SLLN may not be true when only uniform 1st moment for sublinear expectation is finite.

2 Basic Concepts and Lemmas

We use the notations similar to that of Peng[14].Let(?,F)be a given measurable space.Let H be a subset of all random variables on(?,F)such that all IA∈H,where A ∈F and if X1,X2,···,Xn∈ H,then ?(X1,X2,···,Xn)∈ H for each ? ∈ Cl,Lip(Rn),where Cl,Lip(Rn)denotes the linear space of(local Lipschitz)function ? satisfying for some C>0,m ∈ N depending on ?.We consider H as the space of random variables.

Definition 2.1A sublinear expectation E on H is a functionalsatisfying the following properties:for all X,Y∈H,we have

(a)Monotonicity:If X≥Y,then E[X]≥E[Y].

Remark 2.1By combining(b)and(d)in Definition 2.1,we can easily obtain a basic property of sublinear expectation

The triple(?,H,E)is called a sublinear expectation space.Given a sublinear expectation E,let us denote the conjugate expectation E of E by

It is evident that for all X∈H,E[X]≤E[X].

Definition 2.2A set functionis called a capacity if it satisfies

A capacity V is said to be sub-additive if it satisfies

In this paper we only consider the capacity generated by sublinear expectation.Given a sublinear expectation space(?,H,E),we define a capacity:and also define the conjugate capacity:.It is clear that V is a sub-additive capacity and v(A)=E[IA].

The corresponding Choquet expectation(Choquet integral)CVis defined by

Respectively if we change V to v,we can obtain the definition of Cv.Obviously,if V(or v)is the classical probability,then the Choquet expectation CV[X](or Cv[X])coincides with the classical expectation.

Definition 2.3A sublinear expectationis said to be continuous if it satisfies

Now we give the following continuity properties of E and V and the proofs can be referred to Zhang[16].

Proposition 2.1(1)If E is lower-continuous,then it is countably sub-additive,i.e.,

(3)If E is lower-continuous,then V induced by E is lower-continuous.

Example 2.1Let P be a family of probability measures defined on(?,F).For any random variable X,we define a upper expectation by

Next we show the representation theorem of sublinear expectation introduced by Peng[14]and the proof can be found there.

Proposition 2.2Let(?,H,E)be a sublinear expectation space.

(1)(see[14,Theorem 2.4])There exists a family of finitely additive probability measuresdefined on(?,F)such that for each X ∈ H,

(2)(see[14,Lemma 3.4])For any fixed random variable X∈H,there exists a family of probability measures{μθ}θ∈Θdefined on(R,B(R))such that for each ? ∈ Cl,Lip(R),

Definition 2.4Given a capacity V,a set A is said to be a polar set if V(A)=0.And we call a property holds “quasi-surely” (q.s.)if it holds outside a polar set.

We adopt the following notion of independence and identical distribution for sublinear expectation which is initiated by Peng[14].

Definition 2.5(Independence)Let X=(X1,···,Xm),Xi∈ H and Y=(Y1,···,Yn),Yi∈ H be two random variables on(?,H,E).Y is said to be independent of X,if for each test functionwe havewheneverfor all x andis said to be a sequence of independent random variables,if Xn+1is independent of(X1,···,Xn)for each n ≥ 1.

Definition 2.6(Identical Distribution)Let X1,X2be two n-dimensional random variables defined respectively on sublinear expectation spaces(?1,H1,E1)and(?2,H2,E2).They are called identically distributed if

whenever the sublinear expectations are finite.

Definition 2.7(IID Random Variables)A sequence of random variablesis said to be independent and identically distributed,if Xn+1is independent of(X1,···,Xn)and Xnand X1are identically distributed for each n≥1.

To prove our main results,we need the following basic lemmas for sublinear expectation.The proofs of Lemmas 2.1–2.2 can be found in[4].

Lemma 2.1(Borel-Cantelli Lemma)Letbe a sequence of events in F and V be a capacity induced by lower-continuous sublinear expectation E.Ifthen

Lemma 2.2(Chebyshev’s Inequality)Let f(x)>0 be a nondecreasing function on R.Then for any x,

As we all know,in classical probability theory,and E[|X|]<∞are equivalent.But in sublinear situation,this property may not be true.The next lemma reveals the essential difference and the relation between sublinear expectation and Choquet expectation.

Lemma 2.3Given a sublinear expectation space(?,H,E),E is lower-continuous and V is the induced capacity.Then

Proof(1)By the definition of CV,we have

By Proposition 2.2(2)and noticing thatwe have

For any t≥n,let gtbe a function satisfying that its derivatives of each order are bounded,ififand 0≤gt(x)≤1 for all x.Then we have

Hence

It follows that

(2)By the sub-additivity of E,we have

Taking n large sufficiently,there exists some K such that E[|X|I(|X|>n)]≤K and for this n,there holds nV(|X|≤n)<∞.So we have E[|X|]<∞.

(3)Also by the sub-additivity of E,we have

Lemma 2.4If E[|X|]<∞,then|X|<∞q.s.,i.e.,V(|X|=∞)=0.

Proof

Due to the finiteness of E[|X|],letting i→∞,we have V(|X|=∞)=0.

3 The Strong Law of Large Numbers under Controlled 1st Moment Condition

This section is devoted to state and prove the SLLN for sublinear expectation under controlled 1st moment condition.Before we state the main theorem,we need to prove some lemmas.The next lemma is initiated by Cozman[6].

Lemma 3.1If X satisfies a≤X≤b and E[X]≤0,then for any s>0,

ProofThe result is trivial if a=b or if b<0.Now we consider the case a≤0≤b.By convexity of the exponential function,we have

Replacing x with X and taking integral on both sides of the above inequality,we have

Since(esb?esa)(b?a)>0 and E[X]≤0,we have

By some ordinary calculations,we can obtain

So we have ?(0)= ?′(0)=0 and

Then by Taylor’s theorem,we can obtain

Lemma 3.2Given a sublinear expectation space(?,H,E),E is lower-continuous and V is the induced capacity.Letbe a sequence of independent random variables satisfyingfor each n∈N?andfor someLetThen

ProofBy the lower-continuity of V,we only need to prove that for any ε>0,

By Lemma 2.2 and the independence ofwe have for any λ >0,

By Lemma 3.1 and the fact thatandfor each i≥ 1,we have

By Lemma 2.1 we obtain the result.

Remark 3.1By the method in Lemma 3.2,we can also obtain the SLLN for sublinear expectation for a sequence of uniformly bounded random variables.

The following theorem is the main result of this paper.

Theorem 3.1Given a sublinear expectation space(?,H,E),E is lower-continuous and V is the induced capacity.Letbe a sequence of independent random variable withandfor each n∈N?.Suppose that there is a random variable X satisfyingq.s.for each n∈N?andLet

Then

and

ProofBy the monotonicity and sub-additivity of V,we only need to prove

Then by Lemma 3.2,we have

In addition we have

It follows

By the subadditivity and translation invariance of E,we have

Therefore

So we have

Furthermore by Proposition 2.1,Lemma 2.2 and Lemma 2.4,we have

So we have

Combining this with(3.4),we have

Similarly,considering the sequencewithwe have the following equality

This is equivalent to

Remark 3.2If E coincides with the classical expectation,i.e.,V=v=P andEP[X1],where P is the classical probability,our SLLN reduces to the classical SLLN

Corollary 3.1Given a sublinear expectation space(?,H,E),E is lower-continuous and V is the induced capacity.Letbe a sequence of independent random variables withandfor each n ∈ N?.Suppose that there is a random variable X satisfying|q.s.for any n∈N?and

Then

and

ProofTake.Then Ynsatisfies the conditions of Theorem 3.1 with E[Yn]=0.Then we have

This implies

Similarly taking

we can also obtain

4 The Strong Law of Large Numbers under Uniform 1st Moment Condition

In this section we discuss whether the SLLN is valid under conditions of the finiteness of uniform 1st moments for sublinear expectation.

Zhang[16]proved a SLLN for negatively dependent identically distributed random variables under conditions of finite 1st moment for Choquet expectation.By Lemma 2.3(1)and(3)and Theorem 3.1 in Zhang[16],we can obtain the following theorem.

Theorem 4.1Given a sublinear expectation space(?,H,E),E is lower-continuous and V is the induced capacity.Letbe a sequence of i.i.d random variable withand

(2)Suppose that V is continuous.If

By Lemma 2.3(1)and(2),there holds thatimpliesBut the inverse result may not always be true.In some special cases,the finiteness of sublinear expectation can deduce the finiteness of Choquet expectation.For instance,if P only contains finite elements in it,we define a sublinear expectation

Next we give an example to reveal that there does exist a sequence of random variables satisfying the conditions of Theorem 4.1(2)for some certain constructed sublinear expectation such that the SLLN is not true.

Example 4.1Let

be a family of full spaces,Fibe a family of sets each one of which contains all subsets of ?iand

be countable families of countable probability measures,where P1,P2,···,Pn,···defined on each ?iby

for any n ∈ N?.Thensatisfies that for any i,j ≥ 1,

Define the full space

Define the product σ-algebra on ?

Define the set P of probabilities on measure space(?,F)by

We consider the sublinear expectation defined by upper expectation

Then we have

But

then by Theorem 4.1(2)the SLLN is not valid in this example.