Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 754-758
Research Article
754
Some New Properties of Convergence Uncertain Sequence in Measure
Zainab Hayder Abid AL-Aali
1, Mousa M. Khrajan
2, Yaseen M. Alrajhi
31Department of Mathematics and Computer applications, College of Science, Al-Muthanna University, Iraq
Code 28A20
2Department of Mathematics and Computer applications, College of Science, Al-Muthanna University, Iraq
Code 28A20
3Department of Mathematics and Computer applications, College of Science, Al-Muthanna University, Iraq
Code 28A20
1zhayder49@gmail.com,2 mmkrady@mu.edu.iq ,3Yaseenmerzah@mu.edu.iq
Article History: Received: 11 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published
online: 10 May 2021
Abstract: Some a new properties of convergence uncertain sequence in measure are introduced , further more we have that properties of convergence uncertain sequence in distribution were satisfied by using the relation between convergence sequence in measure and in distribution Also, we verify Kolmogorov inequality and some theorem that related with it. Finely new relation between convergence in mean and convergence in distribution were investigated.
Keyword: Convergence; uncertain variable. Uncertain sequence, uncertain measure
1. Introduction
Liu [2]founded in 2007uncertainty theory and Liu [3] refined it in 2010, an uncertain measure is first idea of uncertainty theory defined as a set function 𝑈: ℱ ⟶ ℛsaisfies he following axioms: (1) (Normality Axiom ):
U
(
)
=
1
. (1)(2) (Monotonicity Axiom ): If
A
1A
2, thenU
(
A
1)
U
(
A
2)
(2) (3) (Self-duality Axiom ):U
(
A
1)
+
U
(
A
1c)
=
1
for anyA
1
ℱ(4) (Countable Subadditivity Axiom ):If
{
A
i}
is countable sequence of events, then.)
(
)
(
1 1
= =
i i i iU
A
A
U
(5) (Product Measure Axiom):The product measure
U
is uncertain measure over the product
- fieldℱ1× ℱ2× … × ℱ𝑛 satisfying(
)
(
)
1 1 i n i n i iMin
U
A
A
U
==
for allA
i
i,andi
=
1
,
2
,
3
,...,
n
.Definition (1-2) [1]
The triple
(
,
ℱ,U
)
beuncertainty space such that
, ℱ,andU
be a nonempty set,
- field and uncertain measure respectively.Definition (1-3) [2]
We say that the measurable function
from an uncertainty space to the set of real numbers
is uncertain variable.Definition (1-4) [3]
We say that the uncertain variables
1,
2,...,
n be independent if)
(
min
))
(
(
1 1 i i i n i n i i nU
U
=
=
, for any Borel
1,
2,...,
n of real numbers
.Definition (1-5) [6]
The expected value
of uncertain variable
defined by
− +
−
=
0 0)
(
)
(
)
(
d
d
, provided that at least
+
)
d
(
0 or
− 0)
(
d
.The variance of
is defined by2
))
(
(
)
(
=
−
. Theorem (1-6) [5]Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 754-758
Research Article
755
Suppose that the uncertain variable
X
. Then for any given numbers
0
andr
0
, we haver r
U
)
(
)
(
(3). Definition (1-7)[7]The uncertain sequence
{
n}
be convergent in measure to uncertain variable
if0
}
)
(
)
(
|
{
lim
−
=
→U
y
ny
y
n , for every
0
. Definition (1-8) [7]The uncertain sequence
{
n}
be convergent in mean to uncertain variable
if0
}
)
(
)
(
|
{
lim
−
=
→y
ny
y
n
. Definition (1-9) [7]We say that the uncertain sequence
{
n}
is convergent to uncertain variable in distribution
if)
(
)
(
lim
=
→ nn for all
at which
(
)
is continuous, and
n ,n
=
1
,
2
,...,
k
be an uncertaintydistribution of uncertain variables
n, =
n
1
,
2
,...,
k
.2. Convergence of independent uncertain variables sequence sum Lemma (2-1) [5]
Suppose that
X
i, =
i
1
,
2
,...,
n
be uncertain variables andp
0
. Thenp n i i p p n i i
n
= =
1 1
(4) Theorem (2-2) [7]Suppose that
{
n}
, and
be an uncertain sequence and uncertain variable respectively. if
n→
(in measure)asn
→
.Then
n→
asn
→
(in distribution).Theorem (2-3)
Suppose that
{
n}
, and
be an uncertain sequence and uncertain variable respectively such that
−
=}
{
1
n nU
(5) Then
n→
asn
→
in measure. Proof: Since{
|
(
)
−
(
)
}
{
{
|
(
)
−
(
)
}
=
n m m my
y
y
y
y
y
From (1), and (2) we have
}
)
(
)
(
|
{
{
}
)
(
)
(
|
{
−
−
=
n m m my
y
U
y
y
y
y
U
}
{
−
=n m mU
Furthermore,
lim
{
|
(
)
−
(
)
}
lim
{
−
}
=
0
= → → n m n n m n
U
y
y
y
U
Corollary(2-4)Suppose that
{
n}
, and
be an uncertain sequence, uncertain variables respectively such that
−
=}
{
1
n nU
(6)Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 754-758
Research Article
756
Then
n→
asn
→
in distribution.Proof:-
From theorem (2-2) and from theorem (2-3) according to (5), we have
n→
asn
→
in distributionTheorem (2-5)
Suppose that
{
n}
, and
be an uncertain sequence, uncertain variables respectively such that}
{
2 1
−
= nn is a finite. Then
n→
asn
→
in measure.Proof: Since
{
|
(
)
−
(
)
}
{
{
|
(
)
−
(
)
}
=
n m m ny
y
y
y
y
y
From (2) that}
)
(
)
(
|
{
{
}
)
(
)
(
|
{
−
−
=
n m m ny
y
U
y
y
y
y
U
From theorem (1-6) according to (3) and lemma (2-1) according to (4)
}
)
(
)
(
|
{
{
}
)
(
)
(
|
{
−
−
=
n m m ny
y
U
y
y
y
y
U
2 2}
|
{
{
=−
m n my
2 2}
{
=−
=
m n m 2 2 2
=−
n m mn
Furthermore ,
−
→{
|
(
)
(
)
}
lim
U
y
ny
y
nlim
0
2 2 2=
−
= → n m m nn
Corollary (2-6)Suppose that
{
n}
,
be an uncertain sequence, and uncertain variables respectively such that Then
−
=}
{
2 1
n n (7) .Then
n→
asn
→
in distribution. Proof:-From theorem (2-2) and from theorem (2-4) according to (6), we have
n→
asn
→
in distribution.Theorem (2-7) (Kolmogrov inequality)
Suppose that
i, =
i
1
,
2
,...,
n
be uncertain variables such that
=
=
n i i nW
1
. If
[
i2]
,i
=
1
,
2
,...,
then{
(
)
}
(
)
1 2 2 1−
= n i i i i n iVar
n
W
W
Max
U
for any number
0
.Proof:
From theorem (1-6) according to (3) and lemma (2-1) according to (4), we have
2 1 2 2 2 1 1
(
)
1
)
(
}
)
(
{
−
−
−
−
n i i i i i n i i i n iW
W
Max
W
W
Max
U
= = ==
−
=
−
n i i i n i i n i i iVar
n
n
E
n
1 2 2 2 1 2 2 2 1 2 2)
(
))
(
(
)
(
Theorem (2-8)Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 754-758
Research Article
757
Suppose that
{
i}
be an uncertain sequence. If
= n i i
Var
1)
(
is a finite. Then((
)
(
))
1
=
−
i i i
(8) convergent in measure. Proof: Since((
)
(
))
1
=
−
i i i
convergent in measure if and only iflim
((
)
−
(
))
=
0
= → i n i i n
in measure ifand only if
lim
(
(
)
0
0
=
−
= + = →
k k n n i i i nU
For any
0
,
−
= + =
0)
(
(
k k n n i i iU
−
=
= + = →
m k k n n i i i mU
0)
(
(
lim
−
=
+ = →
k n n i i i m k mlim
U
Max
0(
(
)
))
(
(
lim
2 2
+ = →
−
n m n i i i mn
)
(
lim
2 2
+ = →=
n m n i i mVar
n
lim
2(
)
0
2=
=
= → n i i nVar
n
,from
(
)
=n i iX
Var
is finite. Theorem (2-9)Suppose that
{
i}
be an uncertain sequence. If
= n i i
Var
1)
(
is a finite. Then((
)
(
))
1
=
−
i i i
converges in distribution. Proof:-From theorem (2-2) and theorem (2-8) according to (8), we have
((
)
(
))
1
=
−
i i i
convergent to uncertain variable
in distribution. Theorem (2-10)Suppose that
{
n},
bean uncertain sequence, and uncertain variables respectively. If
n→
as
→
n
in mean, then
n→
asn
→
in distribution.Proof:
Since
n→
asn
→
in mean means
n→
asn
→
in measure, and from theorem (2-2), thus}
{
n is convergence uncertain sequence in distribution to uncertain variable
.Example (2-11)[4]
If
n→
asn
→
in distribution, then
n↛
asn
→
in meanFor example, suppose that the uncertainty space and
=
{
y
1,
y
2}
withU
{
y
1}
=
U
{
y
2}
=
0
.
5
and
=
−
=
=
2 1,
,
)
(
y
ify
b
y
ify
b
y
j i
,and uncertainty distribution of
i and
be
−
−
=
b
b
b
b
,
1
,
5
.
0
,
0
)
(
, that islim
(
)
=
(
)
→ n n indistribution, put
n=
−
we have
n−
=
2
b
for alln
=
1
,
2
,
3
,...,.
y =
y
1, y
2 . Thus
=
=
−
b ndx
b
E
2 02
1
]
[
so that nb
n[
]
2
lim
−
=
→
.Turkish Journal of Computer and Mathematics Education Vol.12 No.11 (2021), 754-758
Research Article
758
3. ConclusionThe rustles of this paper is to obtain new properties of uncertain variable convergence in measure, also Kolmgorov inequality is verified.
References
1. Abid Al-Aali, Z and Almyaly, A.: Some Inequalites on Chance Measure for Uncertain Random Variable .Journal of Engineering and Applied Science . pp. 9602-9605.13(22), (2018)
2. Ahamadzade H , Sheng Y and Hassantabar Darzi.: Some ruselts of moments of uncertain Random Variables. Iranian Journal of Fuzzy Systems. PP.1-21.14(2),(2017)
3. 3. Gao R and Ahamadzade H.: Further ruselts of convergence of uncertain random sequences . Iranian Journal of Fuzzy Systems. PP.31-42.15(4),(2018)
4. Liu, B.: Uncertainty theory. In : Uncertainty theory. pp. 205-234. Springer, (2007) 5. Liu, B.: Uncertainty theory. In: Uncertainty theory. pp. 1-79. Springer, (2010)
6. You, C.: On the convergence of uncertain sequences. Mathematical and Computer Modelling 49(3-4), 482-487 (2009).
7. Zhang, Z.: Some discussions on uncertain measure. Fuzzy Optimization and Decision Making 10(1), 31-43 (2011). doi:10.1007/s10700-010-9091-0
