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This paper extends laws of large numbers under upper probability to sequences of stochastic processes generated by linear interpolation. This extension characterizes the relation between sequences of stochastic processes and subsets of continuous function space in the framework of upper probability. Limit results for sequences of functional random variables and some useful inequalities are also obtained as applications.

Laws of large numbers are the cornerstones of theory of probability and statistics. As we know, under appropriate assumptions, the well-known strong law of large numbers (SLLN for short) states that for a sequence of random variables

When

Recently, Chen [

This paper is motivated by the problem of limit theorems of sequences of stochastic processes in the framework of nonadditive probabilities and the estimation of expectations of functionals of stock prices with ambiguity. If there is no mean uncertainty, they are trivial. But if there is mean uncertainty, then as the SLLN of random variables under nonadditive probability behaves, limit theorems related to stochastic processes become interesting and different from classical case. Chen [

In this paper we will employ the independence condition of Peng [

The remaining part of this paper is organized as follows. In Section

Let

It is obvious that upper probability

Moreover, we can easily get the following properties which are useful in this paper (see also Chen et al. [

For any sequence of sets

Subadditivity of

Lower continuity of

Upper continuity of

If

We say upper probability

The corresponding pair of upper and lower expectations

Let

Throughout this paper we assume (unless otherwise specified) that

Set

Let

Before investigating the convergence problem of sequence

A set

We first give the following property.

The sequence

For each

Obviously, for each

Then the difference of

From

In addition, for any

In fact, without loss of generality, we assume that

Otherwise if

Hence, from (

The following lemma is very useful in the proofs of our main theorems and its proof is similar as Theorem 3.1 of Hu [

Given a sequence of independent random variables

In this section we will investigate the weak convergence problem of

For any

For any

Note that for any integer

Denoting

Let

In addition, since

Then it follows that

For

Obviously,

Since, for all

Let

In particular, if we assume that

In the previous Sections

Any

From Lemma

Let

Take

Thus this theorem is proved.

From the proof of Theorem

By Theorem

Let

In particular,

From the proof of Theorem 3.1 and Corollary 3.2 of Chen et al. [

Supposing

Let

Especially, if we assume

Take

Then this corollary follows from (

In this subsection we will give some useful examples as applications in inequalities.

Let

Especially for

Observe that

By Corollary

Since for any

Thus, inequality (

For any integer

It is easy to check that

We consider a capital market with ambiguity which is characterized by a set of probabilities, denoted the same as previous sections by

For any

This paper proves that any element of subset

The authors declare that there is no conflict of interests regarding the publication of this paper.

This research was partially supported by the WCU (World Class University) Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-20007) and was also partially supported by the National Natural Science Foundation of China (no. 11231005).