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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 3, JUNE 2016
615
Law of Large Numbers for Uncertain
Random Variables
Kai Yao and Jinwu Gao
Abstract—The law of large numbers in probability theory states
that the average of random variables converges to its expected value
in some sense under some conditions. Sometimes, random factors
and human uncertainty exist simultaneously in complex systems,
and a concept of uncertain random variable has been proposed to
study this type of complex systems. This paper aims to provide a
law of large numbers for uncertain random variables, which states
that the average of uncertain random variables converges in distribution to an uncertain variable. As a byproduct, the convergence
of a sequence of uncertain variables is also studied.
Index Terms—Chance theory, law of large numbers, uncertain
random variable, uncertainty theory.
I. INTRODUCTION
N probability theory, the law of large numbers ensures that
the average of some random variables converges to a real
number in some sense under some conditions. First pointed
out by Cardano in the 16th century, the law of large numbers
has been studied by a number of famous researchers, including
Bernoulli, Poisson, Chebyshev, Markov, Borel, Cantelli, Kolmogorov, and Khinchin. Nowadays, many researchers stay focused on this problem. Baum and Katz [1] studied the convergence rate of the law of large numbers, and Hoffmann-Jørgensen
and Pisier [2] studied the law of large numbers in a Banach space.
Acosta [3] gave some inequalities about the law of large numbers for random vectors, and Uhlig [4] provided an application
of law of large numbers in economics. For recent developments
in the law of large numbers, see [5]–[7].
In order to model the fuzzy phenomena in daily life, Zadeh [8]
proposed a concept of fuzzy set, and Zadeh [9] further founded
a possibility theory. Fuzzy measures could be distinguished into
many classes. Valášková and Struk [10], [11] classified the fuzzy
measures into submeasure, supermeasure, submodular, supermodular, belief, plausibility, possibility, and necessity, and they
showed that these classes are closed under the operations of
distortion functions. Besides, both the Choquet integral and the
Sugeno integral were further studied in [12] and [13]. The law of
large numbers for fuzzy sets was first proposed by Fullér [14],
and it was further generalized by Hong and coauthors [15], [16].
In order to model complex phenomena with both randomness
I
Manuscript received January 6, 2015; revised May 13, 2015 and June 24,
2015; accepted July 30, 2015. Date of publication August 7, 2015; date of current
version May 30, 2016. This work was supported in part by the National Natural
Science Foundation of China under Grant 61403360 and Grant 61374082.
K. Yao is with the School of Economics and Management, University of Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]).
J. Gao is with the School of Information, Renmin University of China, Beijing
100872, China (e-mail: [email protected]).
Digital Object Identifier 10.1109/TFUZZ.2015.2466080
and fuzziness, Kwakernaak [17], [18] proposed a concept of
fuzzy random variables. Then, Kruse [19] and Miyakoshi and
Shimbo [20] proposed some laws of large numbers for fuzzy
random variables. These results were further developed by Inoue [21], Kim [22], Joo and Kim [23], and so on.
Except for randomness and fuzziness, human uncertainty is
another source of indeterminacy. In order to deal with human’s
belief degree that some events occur, an uncertainty theory was
founded by Liu [24] in 2007, and refined by Liu [25] in 2009
based on normality, duality, subadditivity, and product axioms.
Similar to probability theory, a concept of uncertain measure is
to indicate the belief degree of possible events, and a concept
of uncertain variable is to model the quantities under uncertain
status in the framework of uncertainty theory. Many researchers
have contributed a lot in this area over the past years. For example, Peng and Iwamura [26] gave a sufficient and necessary
condition for a real function being an uncertainty distribution
of an uncertain variable. Liu and Ha [27] proposed a formula
to calculate the expected value of a function of some uncertain
variables. Chen and Dai [28] verified that a normal uncertain
variable possesses a maximum entropy given the expected value
and variance.
In a complex system, we may have a large samples on some
components but have no samples on some other components.
For the first class of components, we can obtain the probability
distributions of their lifetimes via statistics, but for the second
class, we can only get experts’ belief degree. Therefore, the
system behaves both randomly and uncertainly and cannot be
dealt with simply by probability theory or uncertainty theory. In
order to model such systems, Liu [29], [30] founded a chance
theory based on probability theory and uncertainty theory in
2013. A concept of chance space was proposed by Liu [29]
as a product of probability space and uncertainty space, and a
chance measure, as a generalization of probability measure and
uncertain measure, was also defined to indicate the possibility
that an uncertain random event happens. Then, a concept of
uncertain random variable was proposed to model the quantities
under uncertain and random condition, and chance distribution,
expected value, and variance were employed to describe an
uncertain random variable afterwards.
In 2013, Liu [30] proposed uncertain random programming
as a spectrum of mathematical programming involving uncertain random variables. Then, uncertain random multiobjective
programming, uncertain random goal programming, and uncertain random multilevel programming were developed by Zhou
et al. [31], Qin [32], and Ke et al. [33], [34], respectively. In
addition, chance theory was also applied to risk analysis (see
[35]), reliability analysis (see [36]), graph and network (see
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 3, JUNE 2016
[37]), propositional logic (see [38]), as well as uncertain random process (see [39] and [40]).
This paper will provide a law of large numbers for uncertain
random variables. The rest of this paper is organized as follows.
Section II will review some basic concepts about uncertain
variable and uncertain random variable. Then, the concept of
convergence in distribution for uncertain random variables is
investigated in Section III. After that, the integral of a time function of an uncertain variable is studied in Section IV, and its uncertainty distribution and expected value are studied. Then, the
law of large numbers for uncertain random variables is proved in
Section V, which shows the average of uncertain random variables converges in distribution to an uncertain variable. Finally,
some remarks are made in Section VI.
II. PRELIMINARY
This section will introduce some basic definitions and theorems about uncertain variables and uncertain random variables.
An uncertainty distribution Φ is said to be regular if its inverse
function Φ−1 (α) exists and is unique for each α ∈ (0, 1). An
uncertain variable with a regular uncertainty distribution is usually called a regular uncertain variable in abbreviation. Inverse
uncertainty distribution plays an important role in the operation
of independent uncertain variables.
Definition 5 (see[25]): The uncertain variables ξ1 , ξ2 , . . . ,
ξm on the same uncertainty space are said to be independent
if
m
m
M
(ξi ∈ Bi ) =
M{ξi ∈ Bi }
i=1
i=1
for any Borel sets B1 , B2 , . . . , Bm of real numbers.
Theorem 1 (see [41]): Let ξ1 , ξ2 , . . . , ξn be independent
uncertain variables with regular uncertainty distributions
Φ1 , Φ2 , . . . , Φn , respectively. If the function f (x1 , x2 , . . . , xn )
is strictly increasing with respect to x1 , x2 , . . . , xm and strictly
decreasing with respect to xm +1 , xm +2 , . . . , xn , then
ξ = f (ξ1 , ξ2 , . . . , ξn )
A. Uncertain Variable
Definition 1 (see[24]): Let L be a σ-algebra on a nonempty
set Γ. A set function M : L → [0, 1] is called an uncertain
measure if it satisfies the following axioms:
Axiom 1 (normality axiom): M{Γ} = 1 for the universal set
Γ.
Axiom 2 (duality axiom): M{Λ} + M{Λc } = 1 for any
event Λ.
Axiom 3 (subadditivity axiom): For every countable sequence
of events Λ1 , Λ2 , . . . , we have
∞ ∞
M
Λi ≤
M {Λi } .
i=1
i=1
Then, the triple (Γ, L, M) is called an uncertainty space.
Definition 2 (see [25]): Let (Γk , Lk , Mk ) be uncertainty
spaces for k = 1, 2, . . .. Then, the measure M on the product
σ-algebra is called the product uncertain measure if it satisfies
∞
∞
M
Λk =
Mk {Λk }
k =1
k =1
where Λk are arbitrarily chosen events from Lk for k = 1, 2, . . .,
respectively.
Definition 3 (see [24]): An uncertain variable ξ is a measurable function from Γ to the set of real numbers , i.e., for any
Borel set B of real numbers, the set
{ξ ∈ B} = {γ ∈ Γ | ξ(γ) ∈ B}
is an event.
In order to describe an uncertain variable, a concept of uncertainty distribution is defined as follows.
Definition 4 (see [24]): Let ξ be an uncertain variable. Then,
its uncertainty distribution is defined by
Φ(x) = M{ξ ≤ x}
for any real number x.
is an uncertain variable with inverse uncertainty distribution
−1
Φ−1 (α) = f Φ−1
1 (α), . . . , Φm (α),
−1
Φ−1
m +1 (1 − α), . . . , Φn (1 − α) .
Definition 6 (see [24]): Let ξ be an uncertain variable. Then,
the expected value of ξ is defined by
0
+∞
M{ξ ≥ x}dx −
M{ξ ≤ x}dx
E[ξ] =
−∞
0
provided that at least one of the two integrals is finite.
In order to calculate the expected value of a function of an
uncertain variable, Liu and Ha [27] provided the following theorem.
Theorem 2 (see [27]): Let ξ be an uncertain variable with an
uncertainty distribution Φ. If f (x) is a monotone function such
that E[f (ξ)] exists, then
+∞
f (x)dΦ(x).
E[f (ξ)] =
−∞
B. Uncertain Random Variable
Let (Γ, L, M) be an uncertainty space, and (Ω, A, Pr) be a
probability space. Then
(Γ, L, M) × (Ω, A, Pr) = (Γ × Ω, L × A, M × Pr)
is called a chance space.
Definition 7 (see [29]): Let (Γ, L, M) × (Ω, A, Pr) be a
chance space, and Θ ∈ L × A be an uncertain random event.
Then, the chance measure Ch of Θ is defined by
1
Ch{Θ} =
Pr{ω ∈ Ω | M{γ ∈ Γ | (γ, ω) ∈ Θ} ≥ r}dr.
0
Liu [29] verified that the chance measure Ch satisfies
normality, duality, and monotonicity properties, that is, 1)
Ch{Γ × Ω} = 1; 2) Ch{Θ} + Ch{Θc } = 1 for any event Θ;
and 3) Ch{Θ1 } ≤ Ch{Θ2 } for any events Θ1 and Θ2 with
YAO AND GAO: LAW OF LARGE NUMBERS FOR UNCERTAIN RANDOM VARIABLES
Θ1 ⊂ Θ2 . Besides, Hou [42] proved the subadditivity of chance
measure, that is,
∞
∞
Θi ≤
Ch{Θi }
Ch
i=1
i=1
for a sequence of uncertain random events Θ1 , Θ2 , . . ..
Definition 8 (see [29]): An uncertain random variable ξ is
a measurable function from a chance space (Γ, L, M) ×
(Ω, A, Pr) to the set of real numbers, i.e.,
{ξ ∈ B} = {(γ, ω) | ξ(γ, ω) ∈ B}
is an uncertain random event for any Borel set B.
Random variable and uncertain variable can be regarded as
special cases of uncertain random variable. Let η be a random
variable, τ be an uncertain variable, and f be a measurable
function. Then, f (η, τ ) is an uncertain random variable.
Definition 9 (see [29]): Let ξ be an uncertain random variable. Then, its chance distribution is defined by
Φ(x) = Ch {ξ ≤ x}
∀x ∈ .
The chance distribution of a random variable is just its probability distribution, and the chance distribution of an uncertain
variable is just its uncertainty distribution. For a random variable η with a probability distribution Ψ, an uncertain variable τ ,
and a measurable function f , Liu [30] proved that f (η, τ ) has a
chance distribution
+∞
F (x, y)dΨ(x)
Φ(y) =
−∞
where F (x, y) = M{f (x, τ ) ≤ y} is the uncertainty distribution of f (x, τ ).
Definition 10 (see[29]): Let ξ be an uncertain random variable. Then, its expected value is defined by
0
+∞
Ch{ξ ≥ r}dr −
Ch{ξ ≤ r}dr
E[ξ] =
−∞
0
provided that at least one of the two integrals is finite.
If the uncertain random variable ξ has a chance distribution
Φ, then
+∞
0
E[ξ] =
(1 − Φ(r))dr −
Φ(r)dr.
−∞
0
For a random variable η and an uncertain variable τ , Liu [30]
proved that E[η + τ ] = E[η] + E[τ ] and E[η × τ ] = E[η] ×
E[τ ].
III. CONVERGENCE IN DISTRIBUTION
This section introduces a concept of convergence in distribution for a sequence of uncertain random variables and illustrates
the concept via some examples.
Definition 11: Let ξ, ξ1 , ξ2 , . . . be a sequence of uncertain
random variables with chance distributions Φ, Φ1 , Φ2 , . . ., respectively. Then, {ξi } is said to converge in distribution to ξ
if
lim Φi (x) = Φ(x)
i→∞
617
for every x ∈ at which Φ(x) is continuous.
Remark 1: The concept of convergence in distribution for
uncertain random variables coincides with the concepts of convergence in distribution for random variables and for uncertain
variables, respectively. If {ηi } is a sequence of random variables
converging in distribution to η, then {ηi } is also a sequence of
uncertain random variables that converges in distribution to η.
If {τi } is a sequence of uncertain variables converging in distribution to τ , then {τi } is also a sequence of uncertain random
variables that converges in distribution to τ.
Example 1: Let ξi be an uncertain random variable with a
chance distribution
⎧
0,
if x < 1 − 1/i
⎪
⎪
⎪
⎨
ix − i + 1
Φi (x) =
, if 1 − 1/i ≤ x ≤ 2 + 1/i
⎪
i+2
⎪
⎪
⎩
1,
if x > 2 + 1/i
for i = 1, 2, . . ., and let ξ be an uncertain random variable with
a chance distribution
⎧
0,
if x < 1
⎪
⎪
⎨
Φ(x) = x − 1, if 1 ≤ x ≤ 2
⎪
⎪
⎩
1,
if x > 2.
Since Φi (x) → Φ(x) for every real number x as i → ∞, the
uncertain random sequence {ξi } converges in distribution to ξ.
Lemma 1: Let {τi } be a sequence of uncertain variables converging in distribution to a regular uncertain
variable τ . If f is a continuous and strictly monotone function,
then the uncertain sequence {f (τi )} converges in distribution
to f (τ ).
Proof: The proof breaks into two cases according to the
monotonicity of f . Case 1: Assume f (x) is a strictly increasing
function with respect to x. For simplicity, we define an inverse
function
⎧
−∞, if z ≤ inf{f (x)|x ∈ }
⎪
⎪
⎨
−1
if z = f (y)
f (z) = y,
⎪
⎪
⎩
+∞, if z ≥ sup{f (x)|x ∈ }.
Let Φ denote the uncertainty distribution of τ , and Φi denote the
uncertainty distributions of τi for i = 1, 2, . . .. Then, we have
M{f (τ ) ≤ z} = M{τ ≤ f −1 (z)} = Φ f −1 (z)
M{f (τi ) ≤ z} = M{τi ≤ f −1 (z)} = Φi f −1 (z) .
Since the function
Φ isa regular
uncertainty distribution, we
have Φi f −1 (z) → Φ f −1 (z) as i → ∞ for each real number z. Hence, the sequence {f (τi )} converges in distribution to
f (τ ). Case 2: Assume that f (x) is a strictly decreasing function
with respect to x. Then, −f (x) is a strictly increasing function, and the sequence {−f (τi )} converges in distribution to
−f (τ ) as i → ∞ according to Case 1. As a result, the sequence
{f (τi )} converges in distribution to f (τ ) as i → ∞. The proof
is complete.
Theorem 3: Let {ηi } be a sequence of random variables converging in distribution to η, and {τi } be a sequence of uncertain
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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 3, JUNE 2016
variables converging in distribution to a regular uncertain variable τ . Assume that f (x, y) is a continuous function, and it is
strictly monotone with respect to y for every real number x.
Then, the uncertain random variables
f (ηi , τi ) → f (η, τ )
as i → ∞ in the sense of convergence in distribution.
Proof: Let F (x, y) = M{f (x, τ ) ≤ y} denote the uncertainty distribution of f (x, τ ), and Fi (x, y) = M{f (x, τi ) ≤ y}
denote the uncertainty distributions of f (x, τi ) for i = 1, 2, . . ..
Then, according to Lemma 1, for any real numbers x and y,
we have Fi (x, y) → F (x, y) as i → ∞. Let Ψ denote the probability distribution of η, and Ψi denote the probability distributions of ηi for i = 1, 2, . . .. Then, according to the definition
of convergence in distribution for random variables, we have
Ψi (x) → Ψ(x) as i → ∞ for every x at which Ψ(x) is continuous. Note that f (η, τ ) has a chance distribution
+∞
F (x, y)dΨ(x) = E[F (η, y)]
Ch{f (η, τ ) ≤ y} =
for any given real number y.
Proof: Since the function f (x, y) is strictly increasing with
respect to y for every real number x, the function
+∞
f (x, y)dΦ(x)
−∞
is also increasing with respect to y. Then, on the one hand, for
any γ ∈ Γ such that τ (γ) ≤ y, we have
+∞
+∞
f (x, τ (γ))dΦ(x) ≤
f (x, y)dΦ(x).
−∞
−∞
As a result, we get
+∞
f (x, τ )dΦ(x) ≤
−∞
+∞
−∞
which implies
+∞
M
f (x, τ )dΦ(x) ≤
−∞
−∞
E[Fi (ηi , y)] → E[F (η, y)]
as i → ∞ for each y, the uncertain random sequence {f (ηi , τi )}
converges in distribution to f (η, τ ).
Example 2: Let {ηi } be a sequence of random variables converging in distribution to η, and {τi } be a sequence of uncertain
variables converging in distribution to τ . Then, the uncertain
random variables ηi + τi → η + τ as i → ∞ in the sense of
convergence in distribution.
Example 3: Let {ηi } be a sequence of positive random variables converging in distribution to η, and {τi } be a sequence
of uncertain variables converging in distribution to τ . Then, the
uncertain random variables ηi τi → ητ as i → ∞ in the sense of
convergence in distribution.
Example 4: Let {ηi } be a sequence of random variables
converging in distribution to η, and {τi } be a sequence of
uncertain variables converging in distribution to τ . Then, the
uncertain random variables ηi ∨ τi → η ∨ τ as i → ∞ in the
sense of convergence in distribution, and the uncertain random variables ηi ∧ τi → η ∧ τ as i → ∞ in the sense of
convergence in distribution.
IV. INTEGRAL OF UNCERTAIN VARIABLE
In this section, we introduce the integral of an uncertain variable and obtain its uncertainty distribution and expected value.
Theorem 4: Let τ be an uncertain variable with an uncertainty distribution Ψ, and Φ be a probability distribution of some
random variables. If f (x, y) is a strictly increasing function with
respect to y for every real number x, then
+∞
+∞
f (x, τ )dΦ(x) ≤
f (x, y)dΦ(x) = Ψ(y)
M
−∞
−∞
f (x, y)dΦ(x)
≥ M{τ ≤ y} = Ψ(y).
(1)
On the other hand, for any γ ∈ Γ such that τ (γ) > y, we have
+∞
+∞
f (x, τ (γ))dΦ(x) >
f (x, y)dΦ(x).
−∞
for i = 1, 2, . . .. Since
+∞
−∞
−∞
and f (ηi , τi ) have chance distributions
+∞
Fi (x, y)dΨi (x) = E[Fi (ηi , y)]
Ch{f (ηi , τi ) ≤ y} =
f (x, y)dΦ(x) ⊃ {τ ≤ y}
−∞
As a result, we get
+∞
f (x, τ )dΦ(x) >
−∞
+∞
−∞
f (x, y)dΦ(x) ⊃ {τ > y}
which implies
+∞
f (x, τ )dΦ(x) >
M
−∞
+∞
f (x, y)dΦ(x)
−∞
≥ M{τ > y} = 1 − Ψ(y).
Then
M
+∞
−∞
f (x, τ )dΦ(x) ≤
+∞
f (x, y)dΦ(x)
−∞
≤ M{τ ≤ y} = Ψ(y)
(2)
by the duality of uncertain measure. Combining inequalities (1)
and (2), we have
+∞
+∞
M
f (x, τ )dΦ(x) ≤
f (x, y)dΦ(x) = Ψ(y).
−∞
−∞
Theorem 5: Let τ be an uncertain variable with an uncertainty distribution Ψ, and Φ be a probability distribution of some
random variables. If f (x, y) is a strictly monotone function with
respect to y for every real number x, then
+∞ +∞
+∞
f (x, τ )dΦ(x) =
f (x, y)dΦ(x)dΨ(y).
E
−∞
−∞
−∞
Proof: Since the function f (x, y) is strictly monotone with
respect to y for every real number x, the function
+∞
f (x, y)dΦ(x)
−∞
YAO AND GAO: LAW OF LARGE NUMBERS FOR UNCERTAIN RANDOM VARIABLES
is also monotone with respect to y. According to Theorem 2,
we have
+∞
+∞ +∞
E
f (x, τ )dΦ(x) =
f (x, y)dΦ(x)dΨ(y).
−∞
−∞
−∞
The proof is complete.
V. LAW OF LARGE NUMBERS
In this section, we prove the law of large numbers, which
shows the average of uncertain random variables converges in
distribution to an uncertain variable under some conditions.
Theorem 6 (Law of Large Numbers): Let η1 , η2 , . . . be a sequence of iid random variables with a common probability distribution Φ, and τ1 , τ2 , . . . be a sequence of iid regular uncertain
variables. Define
Sn = f (η1 , τ1 ) + f (η2 , τ2 ) + · · · + f (ηn , τn )
for n ≥ 1, where f (x, y) is a continuous and strictly monotone
function. Then
+∞
Sn
→
f (x, τ1 )dΦ(x)
n
−∞
as n → ∞ in the sense of convergence in distribution.
Proof: The proof breaks into two cases according to the
monotonicity of the function f . Case 1: Assume that f (x, y)
is strictly increasing with respect to y for every x ∈ . Since
f (x, y) is also a continuous function, the function
+∞
f (x, y)dΦ(x)
F (y) =
−∞
is a continuous and strictly
verse function
⎧
−∞,
⎪
⎪
⎨
−1
F (z) = y,
⎪
⎪
⎩
+∞,
increasing function. Define an inif z ≤ inf{F (y)|y ∈ }
if F (y) = z
if z ≥ sup{F (y)|y ∈ }.
Then, we have
+∞
M
f (x, τ1 )dΦ(x) ≤ z
−∞
=M
f (x, τ1 )dΦ(x) ≤
+∞
−∞
= Ψ F −1 (z)
+∞
f x, F
−1
(z) dΦ(x)
−∞
(3)
619
that
f η1 , F −1 (z − ε) + · · · + f ηn , F −1 (z − ε)
≤z
Pr
n
f η1 , F −1 (z − ε) + · · · + f ηn , F −1 (z − ε)
= Pr
n
+∞
−1
≤
f x, F (z − ε) dΦ(x) + ε
−∞
≥1−ε
for any n ≥ N1 . Furthermore, we have
Sn
≤z
n
1 f (η1 , τ1 ) + · · · + f (ηn , τn )
=
≤ z ≥ r dr
Pr M
n
0
1 f (η1 , F −1 (z − ε)) + · · · + f (ηn , F −1 (z − ε))
≤z
≥
Pr
n
0
f (η1 , τ1 ) + · · · + f (ηn , τn )
≤ z ≥ r dr
∩ M
n
1 f (η1 , F −1 (z − ε)) + · · · + f (ηn , F −1 (z − ε))
≤z
≥ Pr
n
0
f (η1 , τ1 ) + · · · + f (ηn , τn )
≤
∩ M
n
f (η1 ,F −1 (z − ε))+· · ·+f (ηn , F −1 (z − ε))
≥ r dr
n
1 f (η1 , F −1 (z − ε)) + · · · + f (ηn , F −1 (z − ε))
≤z
Pr
=
n
0
∩ Ψ F −1 (z − ε) ≥ r dr
Ψ (F −1 (z −ε ) ) f (η1 , F −1 (z − ε))
+ ···
=
Pr
n
0
f (ηn , F −1 (z − ε))
+
≤ z dr
n
−1
≥ Ψ F (z − ε) (1 − ε)
Ch
for any n ≥ N1 . Thus, we obtain that
Sn
≤ z ≥ Ψ F −1 (z)
lim Ch
n →∞
n
(4)
by Theorem 4 where Ψ is an uncertainty distribution of τi .
For
z and any
number
given ε > 0, since
any given real
f η1 , F −1 (z − ε) , f η2 , F −1 (z − ε) , . . . are a sequence of
iid random variables, we have
f η1 , F −1 (z − ε) + · · · + f ηn , F −1 (z − ε)
n
+∞
→
f x, F −1 (z − ε) dΦ(x), a.s.
for any z ∈ . On the other hand, for any given real
number
z and any
given ε > 0, since f η1 , F −1 (z + ε) ,
f η2 , F −1 (z + ε) , . . . are a sequence of iid random variables,
we have
f η1 , F −1 (z + ε) + · · · + f ηn , F −1 (z + ε)
n
+∞
−1
→
f x, F (z + ε) dΦ(x), a.s
as n → ∞ by the strong law of large numbers for random variables. In other words, there exists a positive number N1 such
as n → ∞ by the strong law of large numbers for random variables. In other words, there exists a positive number N2 such
−∞
−∞
620
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 24, NO. 3, JUNE 2016
that
f η1 , F −1 (z + ε) + · · · + f ηn , F −1 (z + ε)
Pr
>z
n
f η1 , F −1 (z + ε) + · · · + f ηn , F −1 (z + ε)
≥ Pr
n
+∞
>
f x, F −1 (z + ε) dΦ(x) − ε
−∞
≥1−ε
for any n ≥ N2 . Furthermore, we have
Sn
>z
n
1 f (η1 , τ1 ) + · · · + f (ηn , τn )
=
> z ≥ r dr
Pr M
n
0
1 −1
f (η1 , F (z + ε)) + · · · + f (ηn , F −1 (z + ε))
>z
≥
Pr
n
0
f (η1 , τ1 ) + · · · + f (ηn , τn )
> z ≥ r dr
∩ M
n
1 f (η1 , F −1 (z + ε)) + · · · + f (ηn , F −1 (z + ε))
>z
≥ Pr
n
0
f (η1 , τ1 ) + · · · + f (ηn , τn )
>
∩ M
n
f (η1 , F −1 (z + ε))+· · ·+f (ηn , F −1 (z + ε))
≥ r dr
n
1 f (η1 , F −1 (z + ε)) + · · · + f (ηn , F −1 (z + ε))
>z
= Pr
n
0
∩ 1 − Ψ F −1 (z + ε) ≥ r dr
1 −Ψ (F −1 (z + ε ) ) f (η1 , F −1 (z + ε))
+ ···
=
Pr
n
0
f (ηn , F −1 (z + ε))
+
> z dr
n
−1
≥ 1 − Ψ F (z + ε) (1 − ε)
Ch
for any n ≥ N2 . By the duality of chance measure, we have
Sn
≤ z ≤ 1 − 1 − Ψ F −1 (z + ε) (1 − ε)
Ch
n
for any n ≥ N2 . Thus, we obtain that
Sn
≤ z ≤ Ψ F −1 (z)
lim Ch
n →∞
n
+∞
−∞
for every point z ∈ , which is apparently equivalent to
+∞
Sn
>z =M
f (x, τ1 )dΦ(x) > z .
lim Ch
n →∞
n
−∞
By the duality of uncertain measure and chance measure, we
further have
+∞
Sn
≤z =M
lim Ch
f (x, τ1 )dΦ(x) ≤ z .
n →∞
n
−∞
The proof is thus complete.
Remark 2: It is believed that Theorem 6 holds in a general
situation: τi is not necessarily regular, and f is not necessarily
continuous. However, a rigorous proof has not been given for
this general case so far.
Example 5: For a sequence of iid random variables
η1 , η2 , . . ., write Sn = η1 + η2 + · · · + ηn for n ≥ 1. Then, it
follows from Theorem 6 that Sn /n converges in distribution
to E[η1 ].
Example 6: For a sequence of iid uncertain variables
τ1 , τ2 , . . ., write Sn = τ1 + τ2 + · · · + τn for n ≥ 1. Then, it
follows from Theorem 6 that Sn /n converges in distribution
to τ1 .
Example 7: Let η1 , η2 , . . . be a sequence of iid random variables, and τ1 , τ2 , . . . be a sequence of iid uncertain variables.
Define Sn = (η1 + τ1 ) + · · · + (ηn + τn ) for n ≥ 1. Then
Sn
→ E[ξ1 ] + τ1
n
in the sense of convergence in distribution as n → ∞.
Example 8: Let η1 , η2 , . . . be a sequence of iid positive random variables, and τ1 , τ2 , . . . be a sequence of iid uncertain
variables. Define Sn = η1 τ1 + · · · + ηn τn for n ≥ 1. Then
Sn
→ E[ξ1 ]τ1
n
in the sense of convergence in distribution as n → ∞.
VI. CONCLUSION
(5)
for any z ∈ . By inequalities (4) and (5), we have
Sn
≤ z = Ψ F −1 (z)
lim Ch
n →∞
n
=M
is a strictly increasing function with respect to y. By a similar
proof to Case 1, we have
+∞
Sn
< −z = M −
f (x, τ1 )dΦ(x) < −z
lim Ch −
n →∞
n
−∞
f (x, τ1 )dΦ(x) ≤ z
and Case 1 is thus proved. Case 2: Assume that f (x, y) is strictly
decreasing with respect to y for every x ∈ . Then, −f (x, y)
Uncertain random variable has been proposed to describe a
system behaving randomly and uncertainly. This paper has proposed a concept of convergence in distribution for a sequence
of uncertain random variables and illustrated the concept via
some examples. Besides, the uncertainty distribution and expected value of the integral of a function of uncertain variable
have also been studied. After that, the law of large numbers that
ensures the average of uncertain random variables converges in
distribution to an uncertain variable under some conditions has
been proved. Based on the law of large numbers, future research
may consider to investigate the renewal processes involving uncertain factors and random factors, or to apply these processes
to replacement policies and insurance risk models as well as the
queuing models.
YAO AND GAO: LAW OF LARGE NUMBERS FOR UNCERTAIN RANDOM VARIABLES
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Kai Yao received the B.S. degree from Nankai University, Tianjin, China, in 2009, and the Ph.D. degree from Tsinghua University, Beijing, China, in
2013.
He is currently an Assistant Professor with
the School of Management, University of Chinese
Academy of Sciences, Beijing, China. He has authored or coauthored more than 30 articles on several
journals including IEEE TRANSACTIONS ON FUZZY
SYSTEMS, Knowledge-Based Systems, Applied Soft
Computing, Applied Mathematical Modelling, Applied Mathematics and Computation, Fuzzy Optimization and Decision Making, and Soft Computing. His current research interests include uncertain renewal processes, uncertain systems, uncertain differential equations, and their
applications.
Jinwu Gao received the B.S. degree in mathematics from Shaanxi Normal University, Xi’an, China,
in 1996, and the M.S. and Ph.D. degrees in mathematics from Tsinghua University, Beijing, China, in
2005.
Dr. Gao is currently an Associate Professor with
the School of Information, Renmin University of
China. His current research interests include fuzzy
systems, uncertain systems, and their application in
optimization, game theory, and finance. He has authored or coauthored more than 30 papers that have
appeared in IEEE TRANSACTIONS ON FUZZY SYSTEMS, Soft Computing, Fuzzy
Optimization and Decision Making, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Iranian Journal of Fuzzy Systems, Computer
& Mathematics with Applications, and other publications.
Dr. Gao has been the Co-Editor-in-Chief of the Journal of Uncertain Systems since 2011 and the Executive-Editor-in-Chief of the Journal of Uncertainty
Analysis and Applications since 2013. He has served as the Vice President and
President of Intelligent Computing Chapter of the Operations Society of China
since 2007 and 2015, respectively, the President of International Consortium
for Electronic Business from 2012 to 2013, the Vice President of International
Consortium for Uncertainty Theory since 2013, and the Vice President of International Association for Information and Management Science from 2010 to
2013.