International Research Journal of Applied and Basic Sciences
© 2014 Available online at www.irjabs.com
ISSN 2251-838X / Vol, 8 (9):1329-1331
Science Explorer Publications
On the Almost Surely Convergence of the Specific
Sequence
Hamid Reza Moradi1, Mona Moradi2
1. Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran
2. Department of Computer Engineering, Ferdows Branch, Islamic Azad University, Ferdows, Iran
Abstract: In this paper, we generalize some results of
random variables with real value, to
random variables with fuzzy value. In fact, we proved that under assumption ∑
sequence
(̃
̃ )
is convergent almost sure to zero, as
(̃ )
, the
.
Here, fuzzy random variables, are pairwise negatively quadrant dependent. We use the Kolmogorov
theorem's for weighted averages of pairwise quadrant random variables.
Key Words: Borel-Cantelli, Fuzzy Random Variables, Negatively Quadrant Dependent, Almostly Sure
Convergence, Chebyshev Inequality, Finite Second Moment, Weighted Sum.
INTRODUCTION
Zadeh in 1965 introduced the concept of fuzzy set by extending the characteristic function [10]. The
occurrence of fuzzy random variable makes the combination of randomness and fuzziness more persuasive, since
the probability theory can be used to model uncertainty and the fuzzy sets theory can be used to model imprecision
E. Deiri and et al [2]. Fuzzy random variables because of its usefulness in several applied fields.
The concept of negative quadrant dependent random variables was introduced by Lehmann [7]. A stronger
notion of negative dependent of the theory of multi-variant probability inequality, Dev and Proschan [3], Szeki [8] is
that of negative association.
In this paper, the authors study some limit properties for a sequence of pairwise
fuzzy random
variables. And extending theorems of negatively dependent real valued random variables to negatively dependent
fuzzy random variables.
Preliminaries
} is said to converge almost surely to (and this denoted as
Definition 1. The sequence{
→
( )
( ))
as
) if (
.
Definition 2. Let be a random variable with finite expected value and finite non-zero variance . Then for any
|
)
real number
, (|
.
( )
)
Lemma 3. Suppose
and ∑
. Then, (
.
In this section, we describe some basic concepts of fuzzy numbers and fuzzy random variables.
[ ], where
Definition 4. Let be a universal set, then a fuzzy set of is defined by its membership function ̃
̃( ) is the membership grade of to ̃.
̃( )
Definition 5. ̃ is called the
set of ̃, defined by ̃
{
}. According to the decomposition
̃
[ ]}. Where ̃ is the indicator function of ordinary set ̃ .
theorem of fuzzy sets, we have, ( )
{ ̃
( )is said to be a fuzzy random variable associated with (
) if and only if
Definition 6. A mapping ̃
̃ ( )}
)
{(
. Where denote
of Borel in .
1
Corresponding Author E-mail: [email protected]
Intl. Res. J. Appl. Basic. Sci. Vol., 8 (9), 1329-1331, 2014
For a fuzzy random variable ̃ and
, let ̃( ) be a fuzzy set with the membership function ̃( )( ).
( )
( ) [
) By the equation,
Definition 7.
( ̃ ̃) {(∫ | (
)
(
)(
)
|)
| (
) (
)(
)|
The analytical properties of
depend on the first parameter , while the second parameter
of
characterizing the subjective weight attributed to the sides of the fuzzy numbers.
) if
Definition 8. A pair of fuzzy random variables(̃ ̃) is called negatively quadrant dependent (
()
()
()
()
̃
[ ]
[̃
]
[̃
] [̃
]
.
Definition 9. A set of fuzzy random variables is called
is for all pairs(̃ ̃) in the set ̃ and ̃ satisfied in
Definition 8.
Definition 10. A pair of fuzzy random variables (̃ ̃) are said to be negatively associated ( ) if
()
()
[ ].
{( (̃)) ( (̃)) }
Theorem 11. Let {
∞.
Let {̃
̃
∞ for
Then, as
Proof. Let
(̃
we have,
and for each
̃ )
→
(̃
{
()
()
{
set,
̃ )
)
(̃
∑
}
}. Since
)
∑
monotonicity of ̃
(̃
,
for
∞ it follows that
as
}
by the Theorem 1 in [5], {
and
by
the
Theorem
(̃ )
∑
̃ )
∑
(̃
(̃
,
(̃ )
(̃
. Thus, by the Borel-Cantelli Lemma,
positive integer. for
every
∞ when
and
.
all large . Therefore, for some
and every
{
}.
By
Chebyshev
inequality
(
such that
(̃ )
∑
∑
∑
be a sequence non-negative and
} be a sequence of negatively quadrant dependence of non negative fuzzy random variables. Set
̃ ). If
[
(̃ ̃ )]
(
I.
II.
}
̃ )
̃
̃ )
.
For
(̃
in
since
. Now, given
(̃
̃ )
}
[4],
every
as
)
{
1
for
[
be a
̃ ). By the
( ̃ ̃ )](
) for
is almost sure. This concludes the proof.
Main Result
∑
Theorem 12. Let { }
be sequence non-negative and
such that
and
∞ when
∞.
Let {̃
} be a sequence of negatively quadrant dependence of fuzzy random variables. Set
̃
̃ ). If
(
I.
[
(̃ ̃ )] ∞
II.
(̃ )
∑
Then, as
∞ for
∞
Proof. The enough we show that
random variables
(̃
we have
(̃
̃ )
̃ )
→
→
. First, note that
. An equivalent condition
(non increasing) function and
.
is that
̃
{( (̃ ))
is a sequence of pairewise
( )
( (̃ ))
( )
fuzzy
}. For all non decreasing
such that the covariance exists. Hence the fuzzy random variables ̃ Ɵ
̃
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Intl. Res. J. Appl. Basic. Sci. Vol., 8 (9), 1329-1331, 2014
̃
̃ Ɵ ̃
̃
are pairwise
and any assume that, for ̃
We consider the sequence {
}
̃ for real , put ( )
{ }. Clearly ( ) is a non decreasing function.
and it's corresponding sum ̃
()
()
̃
Thus, according to proof of Theorem 11, there holds
{ ( )
( ) }
{
} is a sequence
of non positively correlated non negative fuzzy random variables. According to, our assumptions (I) and (II) with
̃ )
Theorem 11 together, imply that
(̃
as
∞. A similar consideration for negative parts, say
̃
̃ and the fact that ̃ Ɵ
̃
̃ complete the proof.
Now we consider the almost sure convergence of weighted average for pairwise
application of Theorem 12.
Theorem 13. Let {̃
and let
S n  in1 X i
[
I.
fuzzy random variables with finite second moments
. Assume
(̃ ̃ )]
(̃ )
( ̃ ̃)
∑
II.
}be a sequence of pairwise
fuzzy random variables as an
Then for
Proof. Let ̃
[
we have,
(̃
̃ and
̃ ̃
(
̃ )→
[
)]
as
.
̃
̃
(
)] and use Theorem 11 Then {̃
} is a of pairwise
fuzzy random variables. Thus, from Theorem 12 we have,
(̃)
∑
( ̃ ̃)]
( ̃ ̃)]
[
∑
∑
[
(
∑
(̃ )
.
fuzzy random variables with finite second moments
(̃ )
( ̃ ̃)][
[
Then for
( ̃ ̃)]
we have,
Proof. Let̃
[
( ̃ ̃)]
[
variables and
∑
(
Let ̃
(
Since ̃
( ̃ ̃)]
[
̃)
Theorem 14. Let {̃
} be a sequence of pairwise
̃ . Assume
and let ̃
I.
[
(̃ ̃ )]
II.
∑
)
(̃
( ̃ ̃)]
[
̃ and
(̃
̃)
((∑
( ̃ ̃)]
[
[
[
( ̃ ̃)]
.
( ̃ ̃)]
( ̃ ̃)]
[
Clearly {̃
̃)
∑
[
( ̃ ̃)]
̃)→
} is a sequence of pairwise
as
.
fuzzy random
̃ )]. Hence, from (II) of Theorem 12 we have,
{
[
( ̃ ̃)]}
∑
̃)
(
̃ ). Then from proof of Theorem 12,
(
̃)
(
[
( ̃ ̃)]
̃ ) and ̃
(̃
[
(̃ )
∑
( ̃ ̃)]
̃ )→
as
.
. According to Theorem 12.
̃ .
REFERENCES
Birkel T. A Note on the Strong Law of large numbers for positively dependent random variables, Statist Probab. Lett, 7 (1980) 13-20.
Deiri E, et al. Extend the Borel-Cantelli Lemma to Sequences of Non-Independent Random Variables, Applied Mathematical Sciences, 4 (13)
(2010) 637-642.
Dev KJ, Proschan F. negative association of random variables, with applications. Ann. Statist, (1983) 286-295.
Etemadi N. On the Laws of Large numbers of nonnegative random variables, j. Multi. Anal, 13 (1983) 187-193.
Etemadi N. Stability of sums of weighted nonnegative random variables, j. Multi. Anal, 13 (1983) 361-365.
Inoue H. A Strong Law of Large Numbers for fuzzy random variables, Fuzzy Sets and Systems, 41 (1991)285-29.
Lehmann EL. Some concept of dependence, Ann. Math. Statist, 37 (1966) 1137-1153.
Smythe RT. Strong laws of large numbers for-dimensional arrays of random variables. Ann. Of Probab.1, 1, (1973)164–170.
Szekli R. Stochastic Ordering and Dependence in Applied Probability, Lecture Notes in Statistics 97, Springer –Verlag, (1995).
Zadeh LA. Fuzzy sets, inform. And control, 8 (1965) 383-353.
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