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
![](https://www.scirp.org/html/8-7401090\2ccd83cc-a77d-4b62-b669-e9b6dc13f59f.jpg)
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
![](https://www.scirp.org/html/8-7401090\2212e954-bd45-4b15-9aee-062af88f844d.jpg)
a3)
is independent sequence of random variables under
where
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 ![](https://www.scirp.org/html/8-7401090\a1519bee-1a13-4c2f-b258-581317028702.jpg)
for some appropriate events
with ![](https://www.scirp.org/html/8-7401090\ebd42ccc-e14c-454f-a150-0c00073aea9f.jpg)
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
![](https://www.scirp.org/html/8-7401090\fabe9241-c17b-42fa-b246-2b9a76422fdd.jpg)
where P is a family of probabilities on
and weakly compact. Thus its conjugate capacity (see Choquet (1954) [9]), i.e., lower probability is
![](https://www.scirp.org/html/8-7401090\53b7a8c1-9006-4327-a6c5-1d1aaa871a9c.jpg)
where
is the complementary set of A. From Huber and Strassen (1973) [10] V and v also satisfy the following properties.
Proposition 1.
1) ![](https://www.scirp.org/html/8-7401090\9fa71166-ea5e-41e8-9a71-0276c16ae3b9.jpg)
2) ![](https://www.scirp.org/html/8-7401090\bb1dc00a-a6ab-49e7-a6fc-8894c9151511.jpg)
3) ![](https://www.scirp.org/html/8-7401090\d15ebbc0-6cb7-4832-aa6d-13bab25f4c07.jpg)
4)
for all ![](https://www.scirp.org/html/8-7401090\0cf8ad53-989e-4262-a052-f1a921af1129.jpg)
5) lower-continuity of
for all sets in
: if
,
then ![](https://www.scirp.org/html/8-7401090\e8f7a2be-2aeb-4621-9168-6443ddac62ad.jpg)
6) upper-continuity of V for all closed sets: if
closed,
then ![](https://www.scirp.org/html/8-7401090\0740dc69-d68b-4e1c-a05f-951e7063dac2.jpg)
7) lower-continuity of
for all open sets: if
open, then ![](https://www.scirp.org/html/8-7401090\aa746ce1-a94e-411d-9b53-df59dad2c410.jpg)
8) upper-continuity of
for all sets: if
, then ![](https://www.scirp.org/html/8-7401090\4350bfab-1985-4432-9133-dde3a8307c6c.jpg)
Now we introduce an upper expectation
by
in the following
![](https://www.scirp.org/html/8-7401090\6469f4b5-8a83-46f0-991b-29cd07aee5f5.jpg)
for all
such that
is the linear expectation corresponding to
such that ![](https://www.scirp.org/html/8-7401090\783ba135-ae0d-4a47-9174-31701fb18d4a.jpg)
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 all
1) Monotonicity: ![](https://www.scirp.org/html/8-7401090\f2ea614b-caca-44f3-9503-486cb379ce86.jpg)
2) Constant preserving: ![](https://www.scirp.org/html/8-7401090\284a07d1-435a-4479-ab8b-7c7db6f143da.jpg)
3) Sub-additivity: ![](https://www.scirp.org/html/8-7401090\8b2978ce-5544-46d0-83ff-81d9777adc3c.jpg)
4) Positive homogeneity: ![](https://www.scirp.org/html/8-7401090\2de7f6b3-0de4-40f5-8123-d7850b3bbaf5.jpg)
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
![](https://www.scirp.org/html/8-7401090\4ba00d4c-1716-4f0d-887b-85ab5da7f998.jpg)
where ![](https://www.scirp.org/html/8-7401090\6c4cd618-0471-4bf8-bd6d-bc9763492885.jpg)
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 ![](https://www.scirp.org/html/8-7401090\06c17577-0e14-47e4-8361-2c13a2bc861f.jpg)
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 ![](https://www.scirp.org/html/8-7401090\d50637a0-7ddb-4722-93b2-aed7aff81b4f.jpg)
Lemma 6. (Denis-Hu-Peng (2011) [11] Theorem 2) For any ![](https://www.scirp.org/html/8-7401090\6d6e22f7-e1fc-4f4f-8056-0707899d8a92.jpg)
![](https://www.scirp.org/html/8-7401090\7534c3ca-dc2a-4cc9-bddf-43373e6144eb.jpg)
Remark 7. Lemma 6 implies that for any ![](https://www.scirp.org/html/8-7401090\685a4da1-f336-47cc-80d6-dc7ab05ebab7.jpg)
![](https://www.scirp.org/html/8-7401090\39035b6e-8406-43b6-bdb8-e110b0dc0eda.jpg)
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
![](https://www.scirp.org/html/8-7401090\baabe25b-df26-4c85-bf5d-e057bb28cbc9.jpg)
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 ![](https://www.scirp.org/html/8-7401090\9beeb3a3-2021-4a80-b6cd-2e372b06c5e0.jpg)
such that
and ![](https://www.scirp.org/html/8-7401090\b84f9018-45a2-446b-91b5-7e0d53019ae3.jpg)
2) There exist two real constants
such that
![](https://www.scirp.org/html/8-7401090\5568d1a3-9496-4e3c-bd22-2e8c93e348b4.jpg)
3) ![](https://www.scirp.org/html/8-7401090\fa70ab76-f341-4cdb-a136-43e6caf7486b.jpg)
Set
Then for any continuous function
with linear growth on
, we have
![](https://www.scirp.org/html/8-7401090\312ff5cd-4d6f-45ad-b9fb-bec4fd1d7ce3.jpg)
Lemma 10 (Hu [6] Theorem 3.2 (I)). Let
satisfy all the conditions given in Lemma 9, then
![](https://www.scirp.org/html/8-7401090\34fd5339-8865-4296-9bf9-aa1ceec8b2db.jpg)
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 ![](https://www.scirp.org/html/8-7401090\cd4c9b67-26ce-42e8-993a-acedf492410c.jpg)
for all
and ![](https://www.scirp.org/html/8-7401090\02f2c6c8-8027-4a1d-a53a-99c286a6c736.jpg)
Set
Then
(2)
(3)
Proof. It is obvious that we only need to prove one of the Equations (2) and (3), since on ![](https://www.scirp.org/html/8-7401090\9020dae7-b6cb-48fe-b7d6-e50bec008e47.jpg)
![](https://www.scirp.org/html/8-7401090\588ab33a-dcaa-49d0-8d16-66ffcbe88d3b.jpg)
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
![](https://www.scirp.org/html/8-7401090\7637d8b2-af88-46d2-a7ed-1fbcdb482c6d.jpg)
Hence we only need to prove
![](https://www.scirp.org/html/8-7401090\e72a9817-70d6-48f5-a38d-237cc0d4a5ca.jpg)
For any subsequence
of
, we denote
![](https://www.scirp.org/html/8-7401090\ef132501-84f6-4f3c-8437-ebc0251e1c22.jpg)
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 ![](https://www.scirp.org/html/8-7401090\1dfed9a4-f66a-495d-961c-8ded782b630b.jpg)
![](https://www.scirp.org/html/8-7401090\6449b40b-e7ba-44a8-8936-22a4ce5871db.jpg)
Equivalently, we only need to prove for any fixed
![](https://www.scirp.org/html/8-7401090\3bc82504-6166-405d-81b8-57e709014077.jpg)
![](https://www.scirp.org/html/8-7401090\b3328513-3348-4cd5-8aa7-4fcc90e9b0ec.jpg)
Then it is sufficient to find an increasing subsequence
of
such that for any fixed ![](https://www.scirp.org/html/8-7401090\b7877105-7fb6-448c-9539-f59838543806.jpg)
(4)
Noticing that
(5)
since
are all closed sets and V is uppercontinuous for closed sets.
In addition, for any
,
![](https://www.scirp.org/html/8-7401090\20df506c-38dc-4f77-be42-9aa9c3a12924.jpg)
where
and
![](https://www.scirp.org/html/8-7401090\7e2c201a-cd6d-4638-b8f8-c94997aeaefe.jpg)
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
![](https://www.scirp.org/html/8-7401090\cec51a4a-ecb6-46ec-a27f-f81e967fc65e.jpg)
And then by Lemma 9 for any fixed
and
, if we choose a small constant ![](https://www.scirp.org/html/8-7401090\c31bf05a-9e6d-4c1f-bd8f-b4dfd0e56947.jpg)
then there exists an integer
such that for any
, we have
(6)
where we denote
for all fixed
with ![](https://www.scirp.org/html/8-7401090\fcf2fd11-3940-406b-82c1-0ffa70eb3da0.jpg)
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
![](https://www.scirp.org/html/8-7401090\c3fd926a-8632-4c7b-80a8-b03896e772e2.jpg)
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 ![](https://www.scirp.org/html/8-7401090\502105c1-73b3-49cb-995e-ba85a0f19483.jpg)
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
![](https://www.scirp.org/html/8-7401090\9070df79-6e92-4038-a3aa-4883edd54ec9.jpg)
For any
there exist an increasing subsequence
of
, an integer
and an open set
(by Remark 7) satisfying
![](https://www.scirp.org/html/8-7401090\b3845984-fc29-4064-b28f-487681ab3511.jpg)
such that when
we have
![](https://www.scirp.org/html/8-7401090\c02a2441-1e4c-4ea2-9879-25e734e4bb7f.jpg)
Then
is continuous on
By Lemma 6, for any
, we can find a compact set ![](https://www.scirp.org/html/8-7401090\38994060-79ed-4721-b69d-f9eb67bd09b1.jpg)
with
such that
![](https://www.scirp.org/html/8-7401090\4f017a30-78ca-47c7-91b4-ca19f8ae2efd.jpg)
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
![](https://www.scirp.org/html/8-7401090\1379b9f0-7073-4125-95cb-fea7d6d95c8d.jpg)
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 ![](https://www.scirp.org/html/8-7401090\2c5cea26-f0ba-41f9-a533-59b6d5d2a76b.jpg)
![](https://www.scirp.org/html/8-7401090\a8682265-ead0-4b93-be60-aae7a92b3144.jpg)
where
is a cluster of limit points of a real sequence ![](https://www.scirp.org/html/8-7401090\6791b2e5-8320-4c29-bad1-b9c10ba4a12b.jpg)
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
![](https://www.scirp.org/html/8-7401090\3459488e-fb6c-46c8-8040-f0f141876bb7.jpg)
where
is any constant in ![](https://www.scirp.org/html/8-7401090\3150e4a3-0441-4d5b-8255-21d39ffb516d.jpg)
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
![](https://www.scirp.org/html/8-7401090\617aa7de-2cb2-48bb-9f2a-4438abfdb872.jpg)
where
and
for any
and
is any given constant in
and
![](https://www.scirp.org/html/8-7401090\b7600300-dd61-435d-8fc4-d78a35399d66.jpg)
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 ![](https://www.scirp.org/html/8-7401090\80f94b63-0bfe-4f70-8224-bf73299918e2.jpg)
![](https://www.scirp.org/html/8-7401090\b5547a36-2c87-4922-9ddb-2111b497a5da.jpg)
In particular,
![](https://www.scirp.org/html/8-7401090\de7f5bdc-d91a-48bf-9678-47bbfbf059b4.jpg)
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
![](https://www.scirp.org/html/8-7401090\0a159316-40c3-4f72-9818-b1bee319d774.jpg)
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
![](https://www.scirp.org/html/8-7401090\e15a98ff-ee87-4d95-a691-bfbdc47b3c9a.jpg)
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
![](https://www.scirp.org/html/8-7401090\76e3fa35-aced-4ac5-8b2c-29a2f9e023c6.jpg)
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.