With a new notion of independence of random variables, we establish the nonadditive version of weak law of large numbers (LLN) for the independent and identically distributed (IID) random variables under Choquet expectations induced by 2-alternating capacities. Moreover, we weaken the moment assumptions to the first absolute moment and characterize the approximate distributions of random variables as well. Naturally, our theorem can be viewed as an extension of the classical LLN to the case where the probability is no longer additive.

Ever since the definition of capacity was introduced by Choquet [

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 [

So far, the works closely related to our results are Peng [

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

The paper is organized as follows. In Section

In this section, we give an overview of the definitions and properties concerning capacities.

Let

A set function

Let

The lower Choquet expectation of

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.

Let

monotonicity:

positive homogeneity:

translation invariance:

Motivated by the probability case, we give the definition of independence under Choquet expectations naturally.

Let

Independence: random variable

Identical distribution: random variables

IID sequence: a sequence

Motivated by Peng [

Let

A sequence

A capacity

The following proposition and lemma help us overcome the problem of nonadditivity of Choquet expectations.

Let

Let

Subadditivity:

For any constant

where

Sublinearity:

(1) is from Dennerberg [

In this section, we first introduce three lemmas which we will make use of and then state the main theorem with proof.

Let

Set

We now estimate the term

Let

Let

Applying the Taylor expansion for function

Set the upper Choquet expectation

Indeed, since

Combining (

From (

Taking infimum

Hence,

The proof of the lemma is complete.

Let

where

Consider the following:

Let

(1) Set partial sums

Note that, similar to (

On the other hand, by Lemma

Combining (

(2) Since the following conjugate property, for

The condition

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 [

The author declares that there is no conflict of interests regarding the publication of this paper.

The author thanks Professor Zengjing Chen and Dr. Xiaoyan Chen for their valuable suggestions. This research is supported by the National Natural Science Foundation of China (no. 11231005).