Recently, Wang et al. [Computers and Mathematics with Applications 57 (2009) 12321248.] and Wang and Watada [Information Sciences 179 (2009) 40574069.] studied the renewal process and renewal reward process with fuzzy random interarrival times and rewards under the Tindependence associated with any continuous Archimedean tnorm. But, their main results do not cover the classical theory of the random elementary renewal theorem and random renewal reward theorem when fuzzy random variables degenerate to random variables, and some given assumptions relate to the membership function of the fuzzy variable and the Archimedean tnorm of the results are restrictive. This paper improves the results of Wang and Watada and Wang et al. from a mathematical perspective. We release some assumptions of the results of Wang and Watada and Wang et al. and completely generalize the classical stochastic renewal theorem and renewal rewards theorem.
AMS Mathematics Subject Classification : 60A86.
1. Introduction
The theory of fuzzy sets, introduced by Zadeh
[27
,
28]
, has been widely examined and applied to statistics and the possibility theory in recent years. Since Puri and Ralescu’s
[20]
introduction of the concept of fuzzy random variables, there has been growing interest in fuzzy variables
[14
,
15
,
16
,
18
,
19
,
20
,
22
,
23
,
24
,
25
,
26
,
27
,
28
,
29
,
30]
. A number of studies
[7
,
8
,
9
,
10
,
11
,
19]
have investigated renewal theory in the fuzzy environment. Hwang
[12]
considered the stochastic process for fuzzy random variables and proved a theorem for the fuzzy rate of a fuzzy renewal process. Popova andWu
[19]
proposed a theorem presenting the longrun average fuzzy reward by using the strong law of large numbers. Zhao and Liu
[29]
and Hong
[11]
discussed the renewal process by considering the fuzzy interarrival time and proved that the expected reward per unit time is the expected value of the ratio of the reward spent in one cycle to the length of the cycle. Zhao and Tang
[30]
obtained some properties of a fuzzy random renewal process generated by a sequence of independent and identically distributed fuzzy random interval times based on fuzzy random theory, particularly Blackwell’s theorem for fuzzy random variables. Hong
[10]
considered the convergence of fuzzy random elementary renewal variables and total rewards earned by time t in the sense of the extended Hausdorff metric
d
_{∞}
and applied this method to a sequence of independent and identically distributed fuzzy random variables to prove the fuzzy random elementary renewal theorem and the fuzzy random renewal reward theorem. All of these studies used minnormbased fuzzy operations. In general, we can consider the extension principle realized by the means of some tnorm. Such a generalized extension principle yields different operations for fuzzy numbers, in accordance with different tnorms. Recently, many studies have focused on such tnormbased operations regarding fuzzy numbers and fuzzy random variables
[3
,
4
,
5
,
6
,
7
,
22
,
23
,
24
,
25]
.
Recentlt, Wang et al.
[25]
studied a fuzzy random renewal process in which the interarrival times are assumed to be independent and identically distributed fuzzy random variables, using the extension principle associated with a class of continuous Archimedean triangular norms. They discussed the fuzzy random renewal process based on the obtained limit theorems, and derived a fuzzy random elementary renewal theorem for the longrun expected renewal rate. Wang and Watada
[24]
also studied a renewal reward process with fuzzy random interarrival times and rewards under the Tindependence associated with any continuous Archimedean tnorm. The interarrival times and rewards of the renewal reward process are assumed to be positive fuzzy random variables whose fuzzy realizations are Tindependent fuzzy variables. They derived some limit theorems in mean chance measures for fuzzy random renewal rewards and proved a fuzzy random renewal reward theorem for the longrun expected reward per unit time of the renewal reward process. In this paper, we consider the results of Wang et al.
[25]
and Wang and Watada
[24]
again. There are two main points to be improved. The first point is that when the renewal theorem and renewal rewards theorem obtained in these papers degenerate to the corresponding the classical result in stochastic renewal process, the results do not cover those of classical stochastic renewal and renewal reward theories. The second point is that some conditions relate to the membership function of the fuzzy variable and the Archimedean tnorm of the results are restrictive. This paper improves the results of Wang and Watada
[24]
and Wang et al.
[25]
from a mathematical perspective. As a consequence, our results cover the classical theory of the random elementary renewal theorem and random renewal reward theorem. We also release some conditions relating to the membership function of fuzzy variable and Archimedean tnorm of the results of Wang and Watada
[24]
and Wang et al.
[25]
and completely generalize the classical stochastic renewal theorem and renewal rewards theorem.
2. Preliminary
Let (Γ,
P
(Γ),
Pos
) be a possibility space. As defined in
[27]
, a normal fuzzy variable on the real line,
R
is defined as a function
Y
: Γ →
R
which has a unimodal, upper semicontinuous membership function
μ_{Y}
on the real line such that there exists a unique real number
m
satisfying
μ_{Y}
(
m
) = sup
_{x}μ_{Y}
(
x
) = 1. The number
m
=
m
(
Y
) is called the modal value of
Y
.
An
LR
fuzzy variable
Y
= (
a
,
α
,
β
)
_{LR}
has a membership function from the reals into interval [0, 1] satisfying
where
L
and
R
are nonincreasing and continuous functions from [0, 1] to [0, 1] satisfying
L
(0) =
R
(0) = 1 and
L
(1) =
R
(1) = 0.
Let [
Y
]
_{α}
be the
α
level sets with [
Y
]
_{α}
= {
x
∈
R

μ_{Y}
≥
α
} for
α
∈ (0, 1], and
Y
_{0}
= cl{
x
∈
R

μ_{Y}
> 0}. Let
K
(
R
) denote the class of nonempty compact convex subsets of
R
The linear structure induced by the scalar product and the Minkowski addition is that
for all
A
,
B
∈
K
(
R
), and λ ∈
R
. If
d_{H}
is the Hausdorff metric on
K
(
R
), which for
A
,
B
∈
K
(
R
) is given by
then (
K
(
R
),
d_{H}
) is a complete and separable metric space [diamond1]. The norm of an element of
K
(
R
) is denoted by
For a fuzzy variable
Y
and any subset
D
of the real numbers, the quantity
is considered to measure the necessity of
Y
belonging to
D
(see
[3]
). If
D
is an interval (
a
,
b
), we may also write
Nes
{
a
<
Y
<
b
} instead of
Nes
{
Y

D
}.
The credibility of
Y
belonging to
D
and the expected value
E
[
Y
] (see
[18]
) are defined as
provided that at least one of the two integrals is finite. In particular, if
Y
is a nonnegative fuzzy variable (i.e.,
Cr
{
Y
< 0} = 0), then
E
[
Y
] =
Recall that a triangular norm (tnorm for short) is a function
T
: [0, 1]
^{2}
→ [0, 1] such that for any
x
,
y
,
z
∈ [0, 1] the following four axioms are satisfied
[13]
:

(T1) Commutativity:T(x,y) =T(y,x),

(T2) Associativity:T(x,T(y,z)) =T(T(x,y),z),

(T3) Monotonicity:T(x,y) ≤T(x,z) whenevery≤z,

(T4) Boundary condition:T(x, 1) =x.
The associativity (
T
_{2}
) allows us to extend each tnorm
T
in a unique way to an nary operation in the usual way by induction, defining for each
n
tuple(
x
_{1}
,
x
_{2}
, ⋯ ,
x_{n}
) ∈ [0, 1]
^{n}
A tnorm
T
is said to be Archimedean if
T
(
x
,
x
) <
x
for all
x
∈ (0, 1). It is easy to check that the minimum tnorm is not Archimedean. Moreover, from
[13]
, every continuous Archimedean tnorm
T
can be represented by a continuous and strictly decreasing function
f
: [0, 1] → [0,∞] with
f
(1) = 0 and
for all
x
∈ (0, 1), 1 ≤
i
≤
n
, where
f
^{[−1]}
is the pseudoinverse of
f
, defined by
The function
f
is called the additive generator of
T
.
Example 2.1.
Examples of continuous Archimedean
t
–norms with additive generators are listed as follows
[13]
:
(
a
)
Dombi t
−
norm D
,
p
> 0:
with the additive generator
f
(
x
) = ((1 −
x
)/
x
)
^{p}
.
(
b
) Hamacher
t
− norm H, p ≥ 0:
with the additive generator
f
(
x
) = ln((
p
+ (1 −
p
)
x
)/
x
).
(c)
Sklar t
−
norm S
,
p
> 0:
with the additive generator
f
(
x
) = (1/
p
)(
x
^{−p}
− 1).
(d) Frank
t
−
norm F
,
p
> 0,
p
≠ 1:
with the additive generator
f
(
x
) = log
_{p}
((
p
− 1)/(
p^{x}
− 1)).
(e)
Y ager t
−
norm Y
,
p
> 0:
with the additive generator
f
(
x
) = (1 −
x
)
^{p}
.
Definition 2.2
(
[2]
). Let
T
be a tnorm. A family of fuzzy variables {
Y_{i}
,
i
∈
I
} is called
T
independent if for any subset {
i
_{1}
,
i
_{2}
, ⋯ ,
i_{n}
} ⊂
I
with
n
≥ 2,
for any subsets
B
_{1}
,
B
_{2}
, ⋯ ,
B_{n}
of
R
. We say two families of fuzzy variables {
Y_{i}
,
i
∈
I
} and {
Z_{j}
,
j
∈
J
} are mutually
T
independent if for any {
i
_{1}
,
i
_{2}
, ⋯ ,
i_{n}
} ⊂
I
and {
j
_{1}
,
j
_{2}
, ⋯ ,
j_{m}
} ⊂
J
with
n
,
m
≥ 1, fuzzy vectors {
Y
_{i1}
, ⋯ ,
Y_{in}
} and {
Z
_{j1}
, ⋯ ,
Z_{jm}
} are
T
independent.
For
T
independent fuzzy variables
Y_{k}
, 1 ≤
k
≤
m
with possibility distributions
μ_{k}
, 1 ≤
k
≤
m
, and a function
g
:
R^{m}
→
R
, the possibility distribution of
g
(
Y
_{1}
,
Y
_{2}
, ⋯ ,
Y_{m}
) is determined via the possibility distributions
μ
_{1}
,
μ
_{2}
, ⋯ ,
μ_{m}
as
where
T
can be any general tnorm. This is the (generalized) extension principle associated with tnorm.
For example, the sum
Y
_{1}
+
Y
_{2}
+⋯ +
Y_{n}
and corresponding arithmetic mean (
Y
_{1}
+
Y
_{2}
+⋯ +
Y_{n}
)/
n
are the fuzzy variables as defined by
and
respectively.
Following Full
é
r and Keresztfalvi
[5]
[5]
, if
T
is an arbitrary tnorm and {
Y_{k}
} are normal fuzzy variables, then the equality
holds.
Definition 2.3
(
[15]
). Suppose that (Ω,Σ, Pr) is a probability space,
F_{v}
is a collection of fuzzy variables defined on the possibility space (Γ,
P
(Γ),
Pos
). A fuzzy random variable is a map
ξ
: Ω →
F_{v}
such that for any Borel subset
B
of
R
, Pos{
ξ
(
ω
) ∈
B
} is a measurable function of
ω
.
Suppose
ξ
is a fuzzy random variable on Ω, from the above definition, we know for each
ω
∈ Ω,
ξ
(
ω
) is a fuzzy variable. Further, a fuzzy random variable
ξ
is said to be positive if for almost every
ω
, the fuzzy variable
ξ
(
ω
) is positive almost surely.
To a fuzzy random variable
ξ
on Ω, for each
ω
∈ Ω, the expected value of the fuzzy variable
ξ
(
ω
), denoted by
E
[
ξ
(
ω
)], has been proved to be a random variable. Based on such fact, the expected value of the fuzzy random variable
ξ
is defined as the mathematical expectation of the random variable
E
[
ξ
(
ω
)].
Definition 2.4
(
[15]
). Let
ξ
be a fuzzy random variable given on a probability space (Ω,Σ, Pr). The expected value of
ξ
is defined as
Definition 2.5
(
[16]
). Let
ξ
be a fuzzy random variable, and
B
be a Borel subset of
R
. The mean chance of an event
ξ
∈
B
is defined as
The expected value (2) is equivalent to the following form :
A sequence of fuzzy random variables {
ξ_{n}
} is said to be uniformly and essentially bounded if there are two real numbers
b_{L}
and
b_{U}
such that for each
k
= 1, 2, ⋯ , we have Ch {
b_{L}
≤
ξ_{n}
≤
b_{U}
} = 1. Moreover, we have the following convergence mode for a sequence of fuzzy random variables.
3. Fuzzy random renewal and renewal rewards process
Let
ξ_{n}
,
n
= 1, 2,⋯ be a sequence of fuzzy random variables defined on a probability space (Ω,Σ, Pr). For each
n
, we denote
ξ_{n}
as the interarrival time between the (
n
− 1)th and the nth event. Define
It is clear that
S_{n}
is the time when the nth renewal occurs, and for any given
ω
∈ Ω and integer
n
,
S_{n}
(
ω
) =
ξ
_{1}
(
ω
) +⋯ +
ξ_{n}
(
ω
) is a fuzzy variable. Let
N
(
t
) denote the total number of the events that have occurred by time
t
. Then we have
For any
ω
∈ Ω,
N
(
t
)(
ω
) = max{
n

S_{n}
(
ω
) ≤
t
} is a nonnegative integervalued fuzzy variable on the possibility space (Γ,
P
(Γ),
Pos
). That is,
N
(
t
)(
ω
)(
γ
) is a nonnegative integer for any
γ
∈ Γ,
ω
∈ Ω and
t
> 0. We call
N
(
t
) a fuzzy random renewal variable, and the process {
N
(
t
),
t
> 0} a fuzzy random renewal process.
On the basis of the above renewal process {
N
(
t
),
t
> 0} with fuzzy random interarrival times
ξ_{n}
,
n
≥ 1, suppose each time a renewal occurs we receive a reward which is a fuzzy random variable. We denote
η_{n}
as the reward earned at each time of the nth renewal. Let
C
(
t
) represent the total reward earned by time
t
, then we have
where
N
(
t
) is the fuzzy random renewal variable.
In the paper of Wang and Watada
[24]
, and Wang et al.
[25]
, they discussed fuzzy stochastic renewal theories within the following conditions:
A1.
T
can be any continuous Archimedean tnorm with additive generator
f
. The interarrival times {
ξ_{n}
} and rewards {
η_{n}
} are positive fuzzy random variables, and for almost every
ω
∈ Ω, {
ξ_{n}
(
ω
)} and {
η_{n}
(
ω
)} are
T
independent fuzzy variable sequences, respectively.
A2. Π is a nonnegative unimodal real function with Π(0) = 1 and Π(−
r
) = 0, where
r
can be any positive real number. The support of Π is denoted by
.
A3. The convex hull of composition function
f
◦ Π in the support
of Π satisfies:
for any nonzero
r
∈
.
Their two main results are the following.
Theorem 3.1
(Fuzzy random elementary renewal theorem
[25]
).
Suppose ξ
_{1}
, ξ
_{2}
,⋯
is a sequence of fuzzy random interarrival times where μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
for almost every ω
∈ Ω
k
= 1, 2,⋯
with
Π
_{I}
(−
a
) = 0,
a
> 0,
and N
(
t
)
is the fuzzy random renewal variable. If
{
U_{k}
}
is i.i.d. random variable sequences with a finite expected value and U_{k}
≥
a
+
h almost surely, then we have
Theorem 3.2
(Fuzzy random renewal reward theorem
[24]
).
Suppose
(
ξ
_{1}
,
η
_{1}
), (
ξ
_{2}
,
η
_{2}
),⋯
is a sequence of pairs of fuzzy random interarrival times and rewards, where
{
η_{k}
}
is uniformly essentially bounded, μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
with U_{k}
≥
a
+
h almost surely, and μ
_{ηk(ω)}
(
x
) = Π
_{R}
(
x
−
V_{k}
(
ω
))
, for almost every ω
∈ Ω,
k
= 1, 2, ⋯ ,
N
(
t
)
is the fuzzy random renewal variable, and C
(
t
)
is the total reward. If
{
U_{k}
}
and
{
V_{k}
}
are i.i.d. random variable sequences with finite expected values, respectively, and
{
ξ_{k}
(
ω
)}
and
{
η_{k}
(
ω
)}
are mutually Tindependent for almost every ω
∈ Ω
, then
There are several remarks regarding these results which can be improved.
Remark 3.1.
In the assumption A1. the interarrival times {
ξ_{n}
} and rewards {
η_{n}
} need not be the same
T
independent fuzzy variable sequences, respectively.
Remark 3.2.
The condition
U_{k}
≥
a
+
h
in Theorem 3.1 is strong. If {
ξ_{k}
,
k
≥ 1} degenerates to a sequence of i.i.d. random variables, then the result in Theorem 1 does not cover the elementary renewal theorem in the stochastic case (see
[21]
)
Remark 3.3.
The condition
U_{k}
≥
a
+
h
in Theorem 3.2 is strong. If {(
ξ_{k}
,
η_{k}
),
k
≥ 1} degenerates to a sequence of i.i.d. random variables, then the result in Theorem 3.2 does not cover the renewal reward theorem in the stochastic case (see
[21]
)
We consider the condition A3.
for any nonzero
r
∈
.
Example 3.3.
Let Π be the possibility distribution of a normal fuzzy variable with Π(
x
) =
e
^{−x2}
,
x
∈
R
,
Y_{p}
(
x
,
y
) is a Yager
t
−norm with the additive generator
f
(
x
) = (1−
x
)
^{p}
. Then
f
◦Π(
x
) = (1−
e
^{−x2}
)
^{p}
on
=
R
and hence
= 0,
x
∈
.
Example 3.4.
Let Π be the possibility distribution of a normal fuzzy variable with Π(
x
) =
e
^{−xλ}
, 0 < λ < 1,
x
∈
R
,
H
_{1}
(
x
,
y
) is a Hamacher
t
−norm with the additive generator
f
(
x
) = −
lnx
. Then
f
◦ Π(
x
) = 
x

^{λ}
, 0 < λ < 1 on
=
R
and hence co(
f
◦ Π)
(
x
) = 0,
x
∈
.
Remark 3.4.
As we can see in the Examples 3.3 and 3.4, the condition A3. is strong. This assumption can be released.
Remark 3.5.
The condition that {
η_{k}
} is uniformly essentially bounded in Theorem 2 can be released.
Throughout this paper, we discuss fuzzy stochastic renewal and renewal rewards theories within the following conditions:
H1.
T_{i}
,
i
= 1, 2 can be any continuous Archimedean tnorms with additive generators
f_{i}
,
i
= 1, 2. The interarrival times {
ξ_{n}
} and rewards {
η_{n}
} are positive fuzzy random variables, and for almost every
ω
∈ Ω, {
ξ_{n}
(
ω
)} is
T
_{1}
independent fuzzy variable sequences and {
η_{n}
(
ω
)} is
T
_{2}
independent fuzzy variable sequences, respectively.
H2. Π is a nonnegative unimodal real function with Π(0) = 1, Π(
x
) < 1,
x
≠ 0,
3.1. Fuzzy random elementary renewal theorem
. We first consider the following result.
Proposition 3.5.
et T be a continuous Archimedean tnorms with additive generators f. Let
{
Y_{k}
}
be a sequence of Tindependent fuzzy variables with membership function μ_{k}
(
x
) = Π(
x
−
u_{k}
)
and let S_{n}
=
Y
_{1}
+· · ·+
Y_{n}
,
m
(
S_{n}
) =
u
_{1}
+· · ·+
u_{n}
.
Then, for all ϵ
> 0
we have
Proof
. Define
Then
is an upper semicontinuous nonnegative unimodal real function with
,
r
≠ 0 and Π(
x
) ≤
,
x
∈
R
. Let
be fuzzy variable with membership function
and let
. Since
and, for all
ϵ
> 0
it suffices to prove that
And to show this fact, it is also easy to check that, for any 0 <
α
< 1
Let 0 <
α
< 1,
δ
> 0 be given and
f
(1−
δ
) =
t
_{0}
. Suppose that
and let
H_{α,δ}
= {
k

f
(
α_{k}
) >
t
_{0}
}. Then, the number of
H_{α,δ}
is less than or equal to
N
where
N
is the smallest natural number bigger than
f
(
α
)/
t
_{0}
. We note that if
k
∉
H_{α,δ}
, then
α_{k}
≥ 1 −
δ
and if
k
∈
H_{α,δ}
, then
α_{k}
≥
α
Then, we have from (1)
where
B
(0,
t_{α}
) = {
x
: 
x
 ≤
t_{α}
},
t_{α}
=
= max{−
l_{α}
,
r_{α}
} such that
and the second equality above comes from the convexity of [
A
]
_{α}
. Hence, we have, for large
n
,
The first term goes to 0 because
δ
is arbitrary and
by the assumption H2 and the second term goes to 0 as
n
→ ∞. This completes the proof. □
Following the line of the proof in regarding to the Theorem of Hong
[8]
and using Proposition 3.5, the following result is obtained.
Proposition 3.6.
Let T be a continuous Archimedean tnorms with additive generators f. Let
{
Y_{k}
}
be a sequence of a Tindependent fuzzy variable with the membership function μ_{k}
(
x
) = Π(
x
−
u_{k}
)
and let S_{n}
=
Y
_{1}
+⋯ +
Y_{n}
,
m
(
S_{n}
) =
u
_{1}
+· · ·+
u_{n}. If μ_{k}
≥
a
,
k
= 1, 2,⋯
and
(
u
_{1}
+· · ·+
u_{n}
)/
n
→
u as n
→ ∞
, then we have, for all ϵ
> 0
Note 1.
It is noted that since
the condition
implies
From Note 1, we have the following result:
Proposition 3.7.
Let T be a continuous Archimedean tnorms with additive generators f. Let
{
Y_{k}
}
be a sequence of a Tindependent fuzzy variable with the membership function μ_{k}
(
x
) = Π(
x
−
u_{k}
)
and let S_{n}
=
Y
_{1}
+⋯ +
Y_{n}
,
m
(
S_{n}
) =
u
_{1}
+· · ·+
u_{n}. If μ_{k}
≥
a
,
k
= 1, 2,⋯
and
(
u
_{1}
+· · ·+
u_{n}
)/
n
→
u as n
→ ∞,
then we have, for all ϵ
> 0
The following is a wellknown result of probability theory
[1]
.
Lemma 3.8
(
[1]
).
Let
be independent and identically distributed (i.i.d.) random variables. Then E

X
_{1}
 < ∞
iff n
^{−1}
max
_{1≤k≤n}

X_{k}
 → 0 a.s.
iff
a.s
.
The following result is immediate from Proposition 3.7 and Lemma 3.8.
Proposition 3.9.
Suppose ξ
_{1}
,
ξ
_{2}
,⋯
is a sequence of fuzzy random interarrival times where μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
for almost every ω
∈ Ω
k
= 1, 2,⋯
with
Π
_{I}
(−
a
) = 0,
a
> 0,
and N
(
t
)
is the fuzzy random renewal variable. If
{
U_{k}
}
is i.i.d. random variable sequences with finite expected value and U_{k}
≥
a almost surely, then we have
Lemma 3.10
(
[25]
).
Suppose ξ
_{1}
,
ξ
_{2}
,⋯
is a sequence of fuzzy random interarrival times where μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
for almost every ω
∈ Ω
k
= 1, 2,⋯
with
Π
_{I}
(−
a
) = 0,
a
> 0,
and N
(
t
)
is the fuzzy random renewal variable. If U_{k}
≥
a almost surely, then we have
where M is the smallest integer such that M
≥
rt and ξ is a fuzzy variable with μ_{ξ}
(
x
) = Π
_{I}
(
x
).
Definition 3.11
(
[16]
). A sequence {
ξ_{n}
} of fuzzy variables is said to converge in credibility to a fuzzy variable
ξ
, if for every
ε
> 0,
Theorem 3.12.
Suppose ξ
_{1}
,
ξ
_{2}
,⋯
is a sequence of fuzzy random interarrival times where μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
for almost every ω
∈ Ω
k
= 1, 2,⋯
with
Π
_{I}
(−
a
) = 0,
a
> 0,
and N
(
t
)
is the fuzzy random renewal variable. If
{
U_{k}
}
is a sequences of i.i.d. random variables with finite expected value and U_{k}
≥
a almost surely, then we have
Proof
. According to Proposition 3.9, we know that {
N
(
t
)(
ω
)/
t
} converges in credibility to 1/
E
[
U
_{1}
] almost surely. Then, it follows from Liu
[17
, Theorem 3.59] that for any
r
∈
R
,
Let
E
[
U
_{1}
] −
a
=
ϵ
> 0. Since
almost surely, we note that, for almost all (
ω
), there exists
T
(
ω
) > 0, depending on
ω
such that for
M
≥
T
(
ω
),
. By Lemma 3.10, for
r
> 2/
ϵ
As a consequence, by the Lebesgue dominated convergence theorem, we have
Here, we define an order of two fuzzy variables. Let
A
,
B
be two fuzzy variables. We define
A
≤
B
iff
Cr
{
A
≥
r
} ≤
Cr
{
B
≥
r
} for all real number
r
.
Lemma 3.13.
Suppose ξ
_{1}
,
ξ
_{2}
,⋯
is a sequence of fuzzy random interarrival times where μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
for almost every ω
∈ Ω
k
= 1, 2,⋯
with
Π
_{I}
(−
a
) = 0,
a
> 0,
and N
(
t
)
is the fuzzy random renewal variable. If
{
U_{k}
}
is a sequences of i.i.d. random variables with finite expected value and U_{k}
≥
a almost surely, then we have
Proof
. Since
P
{
U_{k}
=
a
} < 1, there exists
δ
> 0 such that
P
{
U_{k}
≥
a
+
δ
} =
p
> 0 for all
n
= 1, 2, 3,⋯ . Define
and
and
N′
(
t
) be the corresponding quantities for the sequence
. It is obvious that, considering
as degenerated fuzzy random variables such that
and
N′
(
t
) ≥
N
(
t
) for each
t
, and hence we have that
Since the random variables
are independent with a Bernoullian distribution, elementary computations show that
Hence we have,
δ
being fixed,
Theorem 3.14
(
[1]
).
Let
{
X_{n}
}
is a sequence of random variables such that sup_{n}E

X_{n}

^{p}
=
M
< ∞
for some p
> 0.
If X_{n}
→
X in distribution, then for each r
<
p
:
Our first main result is the following which improves Theorem 3.1.
Theorem 3.15
(Fuzzy random elementary renewal theorem).
Suppose ξ
_{1}
,
ξ
_{2}
,⋯
is a sequence of fuzzy random interarrival times where μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
k
= 1, 2,⋯
for almost every ω
∈ Ω,
with
Π
_{I}
(−
a
) = 0,
a
> 0,
and N
(
t
)
is the fuzzy random renewal variable. If
{
U_{k}
}
is a sequence of i.i.d. random variables with finite expected value and U_{k}
≥
a almost surely, then we have
Proof
. By Lemma 3.13, we have
Since Theorem 3.12 implies
an application of Theorem 3.14 with
X_{n}
(
ω
) =
E
[
N
(n)(
ω
)]/
n
and
p
= 2 yield (2) with
t
replaced by
n
in (2), from which (2) itself follows at once. □
Remark 3.6.
If {
ξ_{k}
,
k
≥ 1} degenerates to a sequences of i.i.d. random variables, the result in Theorem 3.15 is just the renewal theorem in the stochastic case (see
[21]
).
3.2. Fuzzy random renewal reward theorem.
A slight modification of the result regarding the Theorem of Hong
[8]
using Lemma 3.8 and Proposition 3.5, gives the following result.
Proposition 3.16.
Suppose
(
ξ
_{1}
,
η
_{1}
), (
ξ
_{2}
,
η
_{2}
),⋯
is a sequence of pairs of fuzzy random interarrival times and rewards, where μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
with
Π
_{I}
(−
a
) = 0,
U_{k}
(
ω
) ≥
a
> 0
almost surely, and μ
_{ηk(ω)}
(
x
) = Π
_{R}
(
x
−
V_{k}
(
ω
)),
k
= 1, 2, ⋯ ,
N
(
t
)
is the fuzzy random renewal variable, and C
(
t
)
is the total reward. If
{
U_{k}
}
and
{
V_{k}
}
are i.i.d. random variable sequences with finite expected values, respectively, then for any ϵ
> 0,
Lemma 3.17.
Suppose
(
ξ
_{1}
,
η
_{1}
), (
ξ
_{2}
,
η
_{2}
),⋯
is a sequence of pairs of fuzzy random interarrival times and rewards, where μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
with
Π
_{I}
(−
a
) = 0,
U_{k}
(
ω
) ≥
a
> 0
almost surely, and μ
_{ηk(ω)}
(
x
) = Π
_{R}
(
x
−
V_{k}
(
ω
)),
k
= 1, 2, ⋯ ,
, with
, N
(
t
)
is the fuzzy random renewal variable, and C
(
t
)
is the total reward. If
{
U_{k}
} and {
V_{k}
}
are i.i.d. random variable sequences with E
[
U
_{1}
] < ∞
and E
[
V
_{1}
] < ∞
, then there exists a function
,
r
≥ 0 and a function
,
r
≤ 0
almost surely such that for big t
> 0,
and
and
Proof
. Let
N′
(
t
) be the renewal random variables defined in Lemma 3. We consider that for 0 <
l
_{1}
< 1/
pδ
=
<
l
_{2}
, and
r
≥ 0,
where
and
Then, by the classical renewal theorem, lim
_{t→∞}
N′
(
t
)(
ω
)/
t
=
almost surely, we have for big
t
> 0,
Nest, we consider that
where
η
is a fuzzy variable with
μ_{η}
(
x
) = Π
_{R}
(
x
). Then, we have for big
t
> 0, for some
ϵ
> 0,
and
By similar argument, we have for big
t
> 0, for some
ϵ
> 0,
and
Theorem 3.18.
Suppose
(
ξ
_{1}
,
η
_{1}
), (
ξ
_{2}
,
η
_{2}
),⋯
is a sequence of pairs of fuzzy random interarrival times and rewards, where
{
η_{k}
}
is μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
with
Π
_{I}
(−
a
) = 0,
U_{k}
(
ω
) ≥
a
> 0
almost surely, and μ
_{ηk(ω)}
(
x
) = Π
_{R}
(
x
−
V_{k}
(
ω
)),
for almost every ω
∈ Ω
with
,
k
= 1, 2, ⋯ ,
N
(
t
)
is the fuzzy random renewal variable, and C
(
t
)
is the total reward. If
{
U_{k}
} and {
V_{k}
}
are i.i.d. random variable sequences with E
[
U
_{1}
] < ∞
and E
[
V
_{1}
] < ∞
, then
Proof
. According to Proposition 3.16, we know that {
C
(
t
)(
ω
)/
t
} converges in credibility to
E
[
V
_{1}
]/
E
[
U
_{1}
] almost surely. It follows from Liu
[17
, Theorem3.59] that for almost all
r
≥ 0,
and for almost all
r
≤ 0,
As a consequence, by Lemma 3.17 and the Lebesgue dominated convergence theorem, we have
Theorem 3.19
(
[1]
).
Let
{
X_{n}
}
be a sequence of random variables such that E

X_{n}
 < ∞,
n
= 1, 2, 3,⋯
and X_{n}
→
X in probability. Then the following two propositions are equivalent:
i)
{
X_{n}
} is uniformly integrable
ii)E

X_{n}
 →
E

X
.
Our second main result is the following which improves Theorem 3.2.
Theorem 3.20
(Fuzzy random renewal reward theorem).
Suppose
(
ξ
_{1}
,
η
_{1}
), (
ξ
_{2}
,
η
_{2}
),⋯
is a sequence of pairs of fuzzy random interarrival times and rewards, where
{
η_{k}
}
is μ
_{ξk(ω)}
(
x
) = Π
_{I}
(
x
−
U_{k}
(
ω
))
with
Π
_{I}
(−
a
) = 0,
U_{k}
(
ω
) ≥
a
> 0
almost surely, and μ
_{ηk(ω)}
(
x
) = Π
_{R}
(
x
−
V_{k}
(
ω
))
, for almost every ω
∈ Ω
with
,
k
= 1, 2, ⋯ ,
N
(
t
)
is the fuzzy random renewal variable, and C
(
t
)
is the total reward. If
{
U_{k}
}
and
{
V_{k}
}
are i.i.d. random variable sequences with E
[
U
_{1}
] < ∞
and E
[
V
_{1}
] < ∞
, then
Proof
. Let
N′
(
t
) is the random renewal variable defined in Lemma 3.13. Let Π
_{R′}
be a function with Π
_{R′}
(
x
) = Π
_{R}
(
x
) if
x
≥ 0 and 0, otherwise, and let
be fuzzy random variables with
μ
_{η′k(ω)}
(
x
) = Π
_{R′}
(
x
−
V_{k}
(
ω
)), for almost every
ω
∈ Ω. Similarly, we define Π
_{R′′}
to be a function with Π
_{R′′}
(
x
) = Π
_{R}
(
x
) if
x
≤ 0 and 0, otherwise, and
to be fuzzy random variables with
μ
_{η′′k(ω)}
(
x
) = Π
_{R′′}
(
x
−
V_{k}
(
ω
)), for almost every
ω
∈ Ω. Then we clearly have
and
It is also noted that
for some positive real numbers
v′
,
v′′
> 0, respectively. Then we have
Similarly, we have
Then, since
,
k
= 1, 2,⋯ is a sequence of pairs of i.i.d. random variables, by the classical stochastic renewal reward theorem, we have
and
Then, by Theorem 3.19, the class of random variables
is uniformly integrable, and hence
are uniformly integrable. Since
{
E
[
C
(
t
)(
ω
)]/
t
} is also uniformly integrable. Since, by Theorem 3.18, we have
and hence by Theorem 3.19, again we have
which completes the proof. □
Remark 3.7.
If {(
ξ_{k}
,
η_{k}
),
k
≥ 1} degenerates to a sequences of pair of i.i.d. random variables, the result in Theorem 3.20 is simply the renewal rewards theorem in stochastic case (see
[21]
).
4. Conclusion
In this paper, we considered the random fuzzy elementary renewal theorem and the random fuzzy renewal reward theorem studied by Wang et al.
[25]
and Wang and Watada
[24]
when fuzzy random variables are
T
independent and improved the results from a mathematical perspective. If the sequence of fuzzy random variables degenerates to a sequence of random variables, the main results are just the renewal theorem and the renewal rewards theorem in the stochastic case (see
[21]
).
BIO
Jae Duck Kim received the B.S., M.S. and Ph. D degrees in mathematics from Myongji University in 2006, 2008 and 2010, respectively. Since 2015, he has been a Professor in BangMok College of Basic Studies, Myongji University, Korea.
Department of Mathematics, Myongji University, Yongin 449728, South Korea.
email: jdkim@mju.ac.kr
Dug Hun Hong received the B.S., M.S. degrees in mathematics from Kyungpook National University, Taegu, Korea and Ph. D degree in mathematics from University of Minesota, Twin City in 1981, 1983 and 1990, respectively. From 1991 to 2003, he worked with department of Statistics and School of Mechanical and Automotive Engineering, Catholic University of Daegu, Daegu, Korea. since 2003, he has been a Professor in Department of Mathematics, Myongji University, Korea. His research interests include general fuzzy theory with application and probability theory.
Department of Mathematics, Myongji University, Yongin 449728, South Korea.
email: dhhong@mju.ac.kr
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