ABSTRACT

Strong law of large numbers is a fundamental theory in probability and statistics. When the measure tool is nonadditive, this law is very different from additive case. In 2010 Chen investigated the strong law of large numbers under upper probability V by assuming V is continuous. This assumption is very strong. Upper probabilities may not be continuous. In this paper we prove the strong law of large numbers for an upper probability without the continuity assumption whereby random variables are quasi-continuous and the upper probability is generated by a weakly compact family of probabilities on a complete and separable metric sample space.

1. Introduction

Strong law of large numbers under nonadditive probabilities is a much important theory in uncertainty theories and has more applications in statistics, risk measures, asset pricings and many other fields. In 1999 Marinacci [1] first investigated the strong law of large numbers for sequences of independent and identically distributed (IID for short) random variables relative to a capacity which is continuous and totally monotone and proved that under regularity condition the limit inferior and limit superior of, where, lie between the two Choquet integrals (submean) and (supermean) induced by this capacity with probability 1 under (that is, quasi surely), and furthermore, if is null-additive, then that limit inferior attains the submean and the limit superior attains the supermean quasi surely, respectively. This is different from the law under probability measure P whereby under suitable conditions, such as for IID sequences, converges to the mathematical expectation of almost surely relative to. In 2005 Maccheroni and Marinacci [2] extended the results of Marinacci [1] for a totally monotone capacity on Polish space whereby the bounded variables are continuous or simple, or the capacity is continuous. But the conditions of these two articles on capacity are too strong and not easy to test. And generally capacities can not uniquely determine the (nonlinear) expectations relative to the capacities. Motivated by robust statistics and limit theories under sublinear expectations given by Peng in 2007, Chen [3] in 2010 investigated the strong law of large numbers for a pair of lower and upper probabilities which are induced by a sublinear expectation (see Peng (2012) [4])

whereby the sequence is IID under (the independence is different from classical case and the one in Marinacci [1], see Peng (2012) [4]). He proved if

for some, then those limit inferior and limit superior lie between another submean and supermean which may not equal the ones given by Choquet integrals (see Chen, Wu and Li (2012) [5] for details). Furthermore, if we futher assume that is continuous, then

(1)

Hu (2012) [6] extends the results of Chen [3] to the sequence of non-identically distributed random variables for the same independence and continuity assumptions. Chen and Wu (2011) [7] extends Chen [3] to more weaker independence condition without identical distribution assumption, and proves if we further assume that

is continuous, for any subsequence of, are pairwise weakly independent under

, and there exist some constants and such that

then (1) still holds.

We can see that for the strong law of large numbers (1) under an upper probability, there are two key conditions: well-defined independence and continuity of the upper probability V. The continuity assumption of V is based on the second Borel-Cantelli lemma to get for certain sequence of measurable events But as we know in general V is not continuous, since for nonclosed nonincreasing sequence of measurable events, even if may not hold (see Xu and Zhang (2010) [8] for an example). Hence a natural question is: if is not continuous, whether does (1) hold? In this paper we will give a confirmative answer. We assume a1) is a complete and separable metric space, F is a -algebra of all Borel subsets of, P is a nonempty subset of which is a family of all probabilities on, and is also weakly compact;

a2) For each is quasi-continuous, and

a3) is independent sequence of random variables underwhere is a sublinear expectation corresponding to. In this paper we successfully proved the strong law of large numbers under assumptions a1)-a3) without the continuity assumption of V by transforming that an event occures with probability 1 under V (that is,) to the problem of its complementary event, i.e., where is the conjugate lower probability of V and proving

for some appropriate events with

by using properties of V and.

This paper is organized as follows. In Section 2 we give some basic concepts and useful lemmas. In Section 3 we mainly prove the strong law of large numbers without continuity assumption of upper probability V for IID and continuous sequences. Section 4 extends results of Section 3 and gets the law for non-identically distributed sequence. Section 5 gives an example.

2. Preliminaries

Let be a separable and complete metric space. is a σ-algebra of all Borel subsets of. We introduce an upper probability by

where P is a family of probabilities on and weakly compact. Thus its conjugate capacity (see Choquet (1954) [9]), i.e., lower probability is

where is the complementary set of A. From Huber and Strassen (1973) [10] V and v also satisfy the following properties.

Proposition 1.

1)

2)

3)

4) for all

5) lower-continuity of for all sets in: if, then

6) upper-continuity of V for all closed sets: if closed, then

7) lower-continuity of for all open sets: if open, then

8) upper-continuity of for all sets: if , then

Now we introduce an upper expectation by in the following

for all such that is the linear expectation corresponding to such that

Then is a sublinear expectation (see Peng [4]) on, where is a set of all real-valued random variables such that that is, satisfies that for all1) Monotonicity:

2) Constant preserving:

3) Sub-additivity:

4) Positive homogeneity:

is called a sublinear expectation space in contrast with probability space. Given , we say if for all For is called its supermean, whereas is called its submean. If , then is said to have mean uncertainty.

In the following we introduce some useful concepts (one can refer to Peng (2010) [4] for details).

Definition 2. An n-dimensional random vector is said to be independent from an m-dimensional random vector under, if for all bounded Lipschitz continuous functions, we have

where

Remark 3. In general Y being independent of X under does not imply X being independent of Y. See Example 3.13 of Chapter I in Peng (2012) [4] as a counterexample.

Definition 4. A sequence on is said to be a sequence of independent random variables under, if for any is independent of under

Definition 5. A real random variable is said to be quasi-continuous (q.c. for short) if for any there exists an open set with such that is continuous on

Lemma 6. (Denis-Hu-Peng (2011) [11] Theorem 2) For any

Remark 7. Lemma 6 implies that for any

The following Borel-Cantelli lemma is obvious (the readers also can refer to Peng [4] or Chen [3]).

Lemma 8 (Borel-Cantelli Lemma). For any sequence of events in, if then

Lemma 9 (Hu [6] Theorem 3.1). Let be a sequence of independent random variables on

. We assume 1) For any there exist real constants

such that and

2) There exist two real constants such that

3)

Set Then for any continuous function with linear growth on, we have

Lemma 10 (Hu [6] Theorem 3.2 (I)). Let satisfy all the conditions given in Lemma 9, then

3. Strong Law of Large Numbers

Theorem 11. Let be an independent and continuous sequence under. We assume there exist two real constants such that

for all and

Set Then

(2)

(3)

Proof. It is obvious that we only need to prove one of the Equations (2) and (3), since on

In the following we will prove the Equation (2). It is trivial for and this theorem obviously holds true in this case from (I) of Theorem 1.1 of Chen and Wu [7] or Lemma 10. Hence we only need to consider By Lemma 10 or (I) of Theorem 1.1 of Chen and Wu [7], we have

Hence we only need to prove

For any subsequence of, we denote

Since is a sequence of continuous random variables, thus and are both closed sets in for all and Thus is an open set in for any and Then by the upper-continuity of (see Proposition 1 (6)) for closed sets in, we only need to prove for any fixed constant

Equivalently, we only need to prove for any fixed

Then it is sufficient to find an increasing subsequence of such that for any fixed

(4)

Noticing that

(5)

since are all closed sets and V is uppercontinuous for closed sets.

In addition, for any,

where and

where is any fixed constant in. It is obvious that for any fixed is a bounded and Lipschitz continuous function on. Thus by the independence assumption we know that

And then by Lemma 9 for any fixed and

, if we choose a small constant

then there exists an integer such that for any, we have

(6)

where we denote for all fixed with

Taking for any we can obtain

(7)

Then letting tend to and then letting m tend to on both sides of inequality (7), by Lemma 9 again, we can get

(8)

Thus from (5) and (8) we can obtain

Therefore, (4) holds true. We complete the whole proof. □

Remark 12. If then from the Theorem 11 we can see that This is just a trivial case for sequences without mean uncertainty.

Corollary 13. Let be a sequence of quasicontinuous random variables and satisfy all other conditions except for the continuity given in Theorem 11, then Theorem 11 still holds.

Proof. Similarly as the arguments in the proof of Theorem 11, we only need to prove

(9)

when

By the assumptions we know that for each and any constant there exists an open subset of with such that is continuous on. Denote then by Borel-Cantelli lemma (see Lemma 8) we can obtain

For any there exist an increasing subsequence of, an integer and an open set (by Remark 7) satisfying

such that when we have

Then is continuous on By Lemma 6, for any, we can find a compact set

with such that

Then we have

(10)

For we define

(11)

Then it is obvious that is a capacity on and satisfies all the properties of given in Proposition 1 where is substituted by. We also denote by the set of all random variables such that

. Thus on, is an independent and continuous sequence. Since is also a complete and separable metric space, by Theorem 11 we have

(12)

Then from (10)-(12) we have

(13)

Letting and tend to 0 in inequality (13) we can derive

which implies (9). We complete the whole proof of this corollary. □

4. Extensions

In Section 3 we get that the submean and the supermean are the inferior and superior limits of the arithmetic average of the first random variables given in Theorem 11, respectively, with probability 1 under the upper probability. In fact, except the two values, any other value is still the limit of some subsequence of, with probability 1 under. We can see it in the following theorem.

Theorem 14. Under assumptions of Theorem 11, we have for any

where is a cluster of limit points of a real sequence

Proof. For and, the result has been obtained in Theorem 11. For the trivial case, it is obvious. Now we consider and any We can notice that

where is any constant in

since is upper-continuous for closed sets. Thus we only need to find an increasing subsequence of such that for any we have

(14)

Following the arguments in the proof of Theorem 11 we can obtain

where and for any and is any given constant in and

Then by using the same arguments as in the proof of Theorem 11 we also can prove that (14) holds true. The whole proof is complete. □

The following corollary is obvious.

Corollary 15. Under the conditions of Theorem 11, for any continuous real function on, we have for all

In particular,

We also can extend Theorem 11, Theorem 14 and Corollary 15 to the sequences with different submeans and supermeans as follows.

Theorem 16. Let be an independent and continuous sequence under and satisfy conditions

(1)-(3) of Lemma 9. Set Then for any we have

Proof. By Lemma 10 and the proofs of Theorem 11 and Theorem 14 we only need to check whether (6) and (8) hold true under our assumptions of this theorem. In fact, from Lemma 9 they are obviously satisfied. Hence this theorem holds. □

From the proof of Corollary 13 and Theorem 16 we can immediately obtain the following corollary.

Corollary 17. Theorem 16 still holds when continuity assumption is substituted by quasi-continuity condition and condition (2) of Lemma 9 is replaced by the following condition:

(2') there exist real constants such that

5. An Example

Let with the supremum norm. Then is a Banach space and compact, thus it is a separable and complete metric space with the distance generated by the norm of the space. Then we can define a G-expectation, a special sublinear expectation (see Peng [4] for details), where is a nonnegative real number less than 1. Then for any bounded and independent sequence with the same submean and supermean in under, by Theorem 11 we have

where V is generated by, since this sequence is a sequence of quasi-continuous random variables and from Denis, Hu and Peng [11] can be represented as supremum of a family of linear expectations corresponding to a family of probabilities which is weakly compact.

6. Acknowledgements

This research is supported by WCU (World Class University) program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (R31-20007). The author gratefully thanks the referees for their careful reading.

REFERENCES