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 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 
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 all
1) 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.