A Strong Law of Large Numbers for Set-Valued Random Variables in Gα Space ()
1. Introduction
We all know that the limit theories are important in probability and statistics. For single-valued case, many beautiful results for limit theory have been obtained. In [1], there are many results of laws of large numbers at different kinds of conditions and different kinds of spaces. With the development of set-valued random theory, the theory of set-valued random variables and their applications have become one of new and active branches in probability theory. And the theory of set-valued random variables has been developed quite extensively (cf. [2]- [7] etc.). In [1], Artstein and Vitale used an embedding theorem to prove a strong law of large numbers for independent and identically distributed set-valued random variables whose basic space is
, and Hiai extended it to separable Banach space
in [8]. Taylor and Inoue proved SLLN's for only independent case in Banach space in [7]. Many other authors such as Giné, Hahn and Zinn [9], Puri and Ralescu [10] discussed SLLN's under different settings for set-valued random variables where the underlying space is a separable Banach space.
In this paper, what we concerned is the SLLN of set-valued independent random variables in
space. Here the geometric conditions are imposed on the Banach spaces to obtain SLLN for set-valued random varia- bles. The results are both the extension of the single-valued’s case and the extension of the set-valued’s case.
This paper is organized as follows. In Section 2, we shall briefly introduce some definitions and basic results of set-valued random variables. In Section 3, we shall prove a strong law of large numbers for set-valued inde- pendent random variables in
space.
2. Preliminaries on Set-Valued Random Variables
Throughout this paper, we assume that
is a nonatomic complete probability space,
is a real separable Banach space,
is the set of nature numbers,
is the family of all nonempty closed subsets of
, and
is the family of all nonempty bounded closed convex subsets of
.
Let
and
be two nonempty subsets of
and let
, the set of all real numbers. We define addi- tion and scalar multiplication as
![](//html.scirp.org/file/57649x18.png)
![](//html.scirp.org/file/57649x19.png)
The Hausdorff metric on
is defined by
![]()
for
. For an
in
, let
. The metric space
is complete , and
is a closed subset of
(cf. [6], Theorems 1.1.2 and 1.1.3). For more general hyper-
spaces, more topological properties of hyperspaces, readers may refer to a good book [11].
For each
, define the support function by
![]()
where
is the dual space of
.
Let
denote the unit sphere of
,
the all continuous functions of
, and the norm is defined as ![]()
The following is the equivalent definition of Hausdorff metric.
For each
,
![]()
A set-valued mapping
is called a set-valued random variable (or a random set, or a multifunction) if, for each open subset
of
,
.
For each set-valued random variable
, the expectation of
, denoted by
, is defined as
![]()
where
is the usual Bochner integral in
, the family of integrable
-valued random variables, and
. This integral was first introduced by Aumann [3], called Aumann integral in literature.
3. Main Results
In this section, we will give the limit theorems for independent set-valued random variables in
space. The following definition and lemma are from [1], which will be used later.
Definition 3.1 A Banach space
is said to satisfy the condition
for some
, if there exists a mapping
such that
(i)
;
(ii)
;
(iii)
for all
and some positive constant
.
Note that Hilbert spaces are
with constant
and identity mapping
.
Lemma 3.1 Let
be a separable Banach space which is
for some
and let
be single-valued independent random elements in
such that
and
for each
then
![]()
where
is the positive constant in (iii).
Theorem 3.1 Let
be a separable Banach space which is
for some
. Let
be a sequence of independent set-valued random variables in
, such that
for each
. If
![]()
where
for
and
for
, then
converges with probability 1 in the sense of
.
Proof. Define
![]()
Note that
for each
and that both
and
are independent se- quences of set-valued random variables. Next, for each
and ![]()
![]()
That means
is a Cauchy sequence and hence
![]()
as
. Since convergence in the mean implied convergence in probability, Ito and Nisio’s result in [12] for independent random elements(rf. Section 4.5) provides that
![]()
Then for
, by triangular inequality we have
![]()
By the completeness of
, we can have
converges almost everywhere in the sense of
.
Since by equivalent definition of Hausdorff metric, we have
![]()
For any fixed
, there exists a sequence
, such that
![]()
Then by dominated convergence theorem, Minkowski inequality and Lemma 3.1, we have
![]()
for each
and
. Thus,
is a Cauchy sequence, and hence converges. Hence, by the similar way as above to prove
converges with probability one in the sense of
. We also can prove that
![]()
with probability one in the sense of
. The result was proved. W
From theorem 3.1, we can easily obtain the following corollary.
Corollary 3.2 Let
be a separable Banach space which is
for some
. Let
be a sequence of independent set-valued random variables in
such that
for each
. If
are continuous and such that
and
are non-decreasing, then for each
the convergence of
![]()
implies that
![]()
converges with probability one in the sense of
.
Proof. Let
![]()
If
, by the non-decreasing property of
, we have
![]()
That is
(4.1)
If
, by the non-decreasing property of
, we have
![]()
That is
(4.2)
Then as the similar proof of theorem 3.1, we can prove both
and
converges with probability one, and the result was obtained. W
Acknowledgements
The research was supported by NSFC(11301015, 11401016, 11171010), BJNS (1132008).