Law of Large Numbers under Choquet Expectations

and Applied Analysis 3 Definition 5. Let C V be the upper Choquet expectation induced by a capacity V on F, X ∈ F. For any real-valued function φ onR with φ(X) ∈ F, the distribution function of X is defined by F X [φ] := C V [φ (X)] , ∀X ∈ F. (8) Definition 6. A sequence {X i } ∞ i=1 of random variables is said to converge in distribution (in law) under upper Choquet expectation C V on F, if for each real-valued function φ on R with φ(X i ) ∈ F, i ≥ 1, the sequence {C V [φ(X i )]} ∞ i=1 converges. 2-Alternating Capacities. A lot of work on capacities has focused on the 2-alternating case, since the 2-alternating capacities can be applied in many fields, for example, in game theory as certain convex games (see Shapley [17]), in Bayesian robustness [18], and so forth. Definition 7. A capacity V is 2-alternating if for all A, B ∈ F, V (A ∪ B) ≤ V (A) + V (B) − V (A ∩ B) . (9) The following proposition and lemma help us overcome the problem of nonadditivity of Choquet expectations. Proposition 8 (see Dennerberg [15]). Let (Ω,F) be a measurable space, and let V be a 2-alternating capacity defined on F. Then V is subadditive:


Introduction
Ever since the definition of capacity was introduced by Choquet [1] in 1954, it has been a heated scientific subject worldwide.Since in many application 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.Meanwhile, capacities, the nonadditive probability measures, seem to be a powerful tool to model the uncertainty when the assumption of additivity is suspect (e.g., Augustin [2], Maccheroni and Marinacci [3], Doob [4], and Schmeidler [5]).
In applied statistics and probabilities, law of large numbers (LLN) is one of the most frequently used results.Recently, many authors have investigated different kinds of LLNs for capacities.From 1999 to 2005, Maccheroni and Marinacci [3,6] introduced the definition of independence of random variables relative to a capacity and then presented a strong LLN for totally monotone capacities.Around 2006, by adopting the partial differential equations (PDE), Peng [7][8][9] proved the LLN with the independence of random variables he defined under sublinear expectations.In 2009, Chareka [10] obtained the LLN for Choquet capacities by converting the nonadditive Choquet integral into the additive Lebesgue-Stieltjes integral.Later, Chen [11] derived a natural extension of the classical Kolmogorov strong LLN to the subadditive case and the related application was given by Chen et al. [12].In 2011, using Chebyshev's inequality and Borel-Cantelli lemma for capacities, Li and Chen [13] provided the LLN for negatively correlated random variables.In 2013, Z. Chen and J. Chen [14] proposed a new proof of maximal distribution theorem and then derived the LLN under sublinear expectations with applications in finance.Most of the LLN literature above focuses on the estimation, or the confidence intervals of the average of random variables on a capacity space, only a few gives information of the distributions of random variables.
So far, the works closely related to our results are Peng [7] and Z. Chen and J. Chen [14] in the sense that the three LLNs are all proved from the perspective of convergence in distribution (in law).However, the substantive differences not only lie in assumptions, such as the definitions of independence and the moment conditions, but also lie in proving techniques: Peng achieves LLN with the help of PDE, while our LLN is proved by Feller's pure probabilistic method, using Taylor's expansions and some basic properties of Choquet expectations.
As we known, the key to prove classical LLN are the additivity of probabilities/expectations as well as the moment conditions of random variables.However, the Choquet expectations happen to be the nonadditive expectations.To overcome this nonadditive problem, we adopt the 2-alternating capacity; thus, the Choquet expectations induced by it turn to be subadditive ones, under which we consider the sequence {  } ∞ =1 of IID random variables converges in distribution.

Abstract and Applied Analysis
In this paper, firstly, we introduce a notion of independence of random variables under Choquet expectations, which is different from [7], then establish the nonadditive version of LLN, and characterize the approximate distribution of (1/) ∑  =1   for large .Moreover, our moment condition is weaker than other Choquet literature's.
The paper is organized as follows.In Section 2, we present some basic concepts associated with the Choquet expectations.One of the most important ones is the new concept of independence we introduced in this paper, under which we establish our results.In Section 3, we state the LLN under Choquet expectations with proof followed by the conclusion remarks in Section 4.

Preliminary
In this section, we give an overview of the definitions and properties concerning capacities.
Let (Ω, F) be a measurable space.Suppose that   (R) is the set of all bounded and continuous real-valued functions on R and  2  (R) is the set of functions in   (R) with bounded, continuous first-and second-order derivatives.
(2) V() ≤ V() whenever  ⊆  and ,  ∈ F. Definition 2. Let  be a random variable on (Ω, F).The upper Choquet expectation (integral) of  induced by a capacity V on F is defined by The lower Choquet expectation of  induced by V is given by which is conjugate to the upper expectation and satisfies For simplicity, we can only consider the upper Choquet expectation in the sequel, since the lower (conjuagte) part can be considered similarly.
Lately, a few papers have discussed the Choquet expectation (integral); however, we only need the following basic properties in this paper.
Proposition 3 (see Dennerberg [15]).Let ,  be two random variables on (Ω, F), and let C  be the upper Choquet expectation induced by a capacity V.Then, one has (1) If  and  are said to be independent for any bounded realvalued function  on R, Then, the independence for additive expectation   can be viewed as It is worth pointing out that the independence above is mutual independence due to Fubini's theorem, but for nonadditive Choquet expectations, the mutuality is no longer true; that is, the second equality in (5) does not hold.Motivated by the probability case, we give the definition of independence under Choquet expectations naturally.Definition 4. Let C  be the upper Choquet expectation, , ,   ∈ F,  ≥ 1.Then we have the following.
(ii) Identical distribution: random variables  and  are said to be identically distributed, if for each realvalued function  with (), () ∈ F, (iii) IID sequence: a sequence {  } ∞ =1 of random variables is called independent and identically distributed (IID), if   and   are identically distributed for each ,  ⩾ 1 and  +1 is independent of ∑  =1   .
Motivated by Peng [7], we give the notions of distribution functions and convergence in distribution for Choquet's case.Definition 5. Let C  be the upper Choquet expectation induced by a capacity V on F,  ∈ F. For any real-valued function  on R with () ∈ F, the distribution function of  is defined by

2-Alternating Capacities.
A lot of work on capacities has focused on the 2-alternating case, since the 2-alternating capacities can be applied in many fields, for example, in game theory as certain convex games (see Shapley [17]), in Bayesian robustness [18], and so forth.
The following proposition and lemma help us overcome the problem of nonadditivity of Choquet expectations.Proposition 8 (see Dennerberg [15]).Let (Ω, F) be a measurable space, and let V be a 2-alternating capacity defined on F. Then V is subadditive: Lemma 9. Let C  be the upper Choquet expectation induced by 2-alternating capacity V.For any ,  ∈ F, one has the following.

Main Results
In this section, we first introduce three lemmas which we will make use of and then state the main theorem with proof.
Lemma 10.Let {  } ∞ =1 be a sequence of independent random variables on (Ω, F).Let V be a 2-alternating capacity defined on F, and let C  , C  be the induced upper, lower Choquet expectations, respectively.Then for any  ∈   (R) and any constant   ∈ R, where Proof. where We now estimate the term This with the sublinearity of C  in Lemma 9, for one side, implies, For the other side, That is, inf This with (14) implies that It then follows by the fact that sup We complete the proof of this lemma.
Combining (31) and (32), for the arbitrary of , as  → ∞, we have (1) From (33), we have (2) Taking infimum inf ∈R on both sides of (28), similar to the proof of (1), we have Hence, The proof of the lemma is complete.
This with (46) implies that lim sup Note that, similar to (32),   () in ( 47) is still a positive constant with   () → 0 as  → ∞.To be specific, due to the following facts: (a) the part for lower Choquet expectation C  can be proved similarly, we just omit it.
Remark 14.The condition  ∈  2  (R) in Theorem 13 can be weaken to  ∈   (R).Further, the IID condition can be weaken to "the independent random variables satisfying the Choquet absolute moment condition"; see our future work.

Conclusion Remarks
In summary, under the new concept of independence, we propose the nonadditive version of LLN under Choquet expectations induced by 2-alternating capacities and characterize the approximate distributions of random variables.Meanwhile, we weaken the moment conditions to the first absolute moment condition compared to other Choquet LLNs' , such as Chareka [10].Our main technique is purely probabilistic and elementary, thus, it can be viewed as a natural extension of the traditional LLN to the case where the probability is no longer additive.