A law of large numbers for fuzzy numbers ∗
Robert Fullér
rfuller@abo.fi
Abstract
We study the following problem: If ξ1 , ξ2 , . . . are fuzzy numbers with modal values M1 , M2 , . . . ,
then what is the strongest t-norm for which
ξ1 + · · · + ξn
lim Nes mn − ≤
≤ mn + = 1,
n→∞
n
for any > 0, where
mn =
M 1 + · · · + Mn
,
n
the arithmetic mean
ξ1 + · · · + ξn
n
is defined via sup-t-norm convolution and Nes denotes necessity.
Keywords: Possibility, probability, necessity, fuzzy number, triangular norm, law of large numbers, sequence of fuzzy numbers, convergence theorem.
1
Definitions
A fuzzy number ξ is a fuzzy set of the real line R with a unimodal, normalized (i.e. there exists unique
a ∈ R such that ξ(a) = 1) and upper-semicontinuous membership function [4].
Given a subset D ⊂ R, the grade of possibility of the statement ”D contains the value of ξ” is defined by
[11]
Pos(ξ | D) = sup ξ(x)
(1)
x∈D
The quantity 1−Pos(ξ | D̄), where D̄ is the complement of D, is denoted by Nes(ξ|D) and is interpreted
as the grade of necessity of the statement ”D contains the value of ξ”.
It satisfies dual property with respect to (1):
Nes(ξ | D) = 1 − Pos(ξ|D̄).
If D = [a, b] ⊂ R then instead of Nes(ξ | [a, b]) we shall write Nes(a ≤ ξ ≤ b) and if D = x, x ∈ R we
write Nes(ξ = x).
∗
The final version of this paper appeared in: R. Fullér, A law of large numbers for fuzzy numbers, Fuzzy Sets and Systems,
45(1992) 299-303. doi: 10.1016/0165-0114(92)90147-V
1
Let ξ1 , ξ2 , . . . be a sequence of fuzzy numbers. We say that {ξn } converges pointwise to a fuzzy set ξ
(and write limn→∞ ξn = ξ) if
lim ξn (x) = ξ(x),
n→∞
for all x ∈ R. Given two fuzzy numbers, ξ and η, their T -sum ξ + η is defined by [3,12]
(ξ + η)(z) = sup T (ξ(x), η(y)), x, y, z ∈ R
x+y=z
where T is a triangular norm [10] (t-norm for short), i.e. T is a two-place function from [0, 1] × [0, 1] →
[0, 1] such that T is symmetric, associative, non-decreasing and T (x, 1) = x, x ∈ [0, 1].
The function Hγ : [0, 1] × [0, 1] → [0, 1], where γ > 0, defined by
Hγ (u, v) =
uv
γ + (1 − γ)(u + v − uv)
is called Hamacher-norm with parameter γ (Hγ -norm for short) [6].
A symmetric triangular fuzzy number ξ denoted by (a, α) is defined as

|a − t|

if |a − t| ≤ α,
1−
ξ(t) =
α

0
otherwise,
where a ∈ R is the modal value and 2α > 0 is the spread of ξ.
Let T1 , T2 be t-norms. We say that T1 is weaker than T2 (and write T1 ≤ T2 ) if T1 (x, y) ≤ T2 (x, y) for
each x, y ∈ [0, 1]. If ξ is a fuzzy number then its support denoted by suppξ is defined as suppξ = {t ∈
R : ξ(t) > 0}.
2
Chebyshev’s form of the law of large numbers
We shall provide a fuzzy analogue of the following theorem [1].Chebyshev’s theorem. If ξ1 , ξ2 ,. . . is a
sequence of pairwise independent random variables having finite variances bounded by the same constant
Dξ1 ≤ C, Dξ2 ≤ C, . . . , Dξn ≤ C, ...
and
M = lim
n→∞
M1 + · · · + Mn
n
exists, then for any positive constant ξ1 + · · · + ξn M1 + · · · + Mn −
lim Prob < =1
n→∞
n
n
where Mn = M ξn and Prob denotes probability.
3
A law of large numbers for fuzzy numbers
In this section we shall prove that if ξ1 = (M1 , α), ξ2 = (M2 , α) . . . is a sequence of symmetric triangular
fuzzy numbers and T is a t-norm (by which the sequence of arithmetic means {(ξ1 + · · · + ξn )/n} is
defined) then
2
(a) the relation
ξ1 + · · · + ξn
lim Nes mn − ≤
≤ mn + = 1, for any > 0
n→∞
n
(2)
holds for any T ≤ H0 ,
(b) the relation (2) is not valid for the ”min”-norm.
Let T be a t-norm and let ξ, ξ2 , . . . be a sequence of fuzzy numbers. We shall say that the sequence {ξn }
obeys the law of large numbers if it satisfies the relation (2).
Lemma 3.1 Let ξ, η be fuzzy sets of R. If ξ ⊆ η (i.e. ξ(x) ≤ η(x), for each x ∈ R) then
Nes(ξ = x) ≥ Nes(η = x), for each x ∈ R.
Proof 1 From the definition of necessity we have
N es(ξ = x) = 1 − P os(ξ|R \ {x}) = 1 − sup ξ(t) ≥ 1 − sup η(t) = N es(η = x),
t6=x
t6=x
which ends the proof.
The proof of the next two lemmas follows from the definition of t-sum of fuzzy numbers.
Lemma 3.2 Let T1 , T2 be t-norms and let ξ1 , ξ2 be fuzzy numbers. If T1 ≤ T2 then
(ξ1 + ξ2 )1 ⊆ (ξ1 + ξ2 )2
where (ξ1 + ξ2 )i denotes the Ti -sum of fuzzy numbers ξ, η, i = 1, 2.
Lemma 3.3 Let ξi = (ai , α), i = 1, . . . , n be fuzzy numbers of symmetric triangular form and let T be
a t-norm. Then
supp(ξ1 + · · · + ξn ) ⊆ suppξ1 + · · · + suppξn =
[a1 − α, a1 + α] + · · · + [an − α, an + α] = [a1 + · · · + an − nα, a1 + · · · + an + nα]
where the sum ξ1 + · · · + ξn is defined via sup-T convolution.
Lemma 3.4 Let T = H0 and ξi = (ai , α), i = 1, 2, . . . , n. Then with the notations
ηn = ξ1 + · · · + ξn ,
An = a1 + · · · + an
we have
(i)
(ii)



1 − |An − z|(nα)−1
if |An − z| ≤ nα,
1 + (n − 1)|An − z|(nα)−1
ηn (z) =


0
otherwise,

!
1 − |An /n − z|α−1


ηn
if |An /n − z| ≤ α,
1 + (n − 1)|An /n − z|α−1
(z) =

n

0
otherwise,
3
Proof 2 We prove (i) by making an induction argument on n. Let n = 2. Then we need to determine the
value of η2 (z) from the following relationship:
η2 (z) = sup
x+y=z
ξ1 (x)ξ2 (y)
= sup
ξ1 (x) + ξ2 (y) − ξ1 (x)ξ2 (y) x+y=z
1
1
1
+
−1
ξ1 (x) ξ2 (y)
,
if z ∈ (a1 + a2 − 2α, a1 + a2 + 2α) and η2 (z) = 0 otherwise.
According to the decomposition rule of fuzzy numbers into two separate parts, η2 (z), z ∈ (a1 + a2 −
2α, a1 + a2 ], is equal to the value of the following mathematical programming problem
1
1
a1 − x
1−
α
+
1
a2 − z + x
1−
α
→ max
(3)
−1
subject to a1 − α < x ≤ a1 ,
a2 − α < z − x ≤ a2 .
Using Lagrange’s multipliers method for the solution of the problem (3) we get that its value is
a1 + a2 − z
A2 − z
1−
2α
2α
=
a1 + a2 − z
A2 − z
1+
1+
2α
2α
1−
and the solution of (3) is
a1 − a2 + z
2
(where the first derivative vanishes). If a1 + a2 ≤ z < a1 + a2 + 2α then we need to solve the following
problem
1
→ max
(4)
1
1
+
−1
x − a1
z − x − a2
1−
1−
α
α
subject to a1 < x < a1 + α,
x=
a2 < z − x < a2 + α.
In a similar manner we get that the value of (4) is
z − A2
2α
z − A2
1+
2α
1−
and the solution of (4) is
x=
a1 − a2 + z
2
4
(where the first derivative vanishes). Let us assume that (i) holds for some n ∈ N . Then,
ηn+1 (z) = (ηn + ξn+1 )(z), z ∈ R,
and by similar arguments it can be shown that (i) holds for ηn+1 . The statement (ii) can be proved
directly using the relationship (ηn /n)(z) = ηn (nz), z ∈ R. This ends the proof.
The following lemma shows that if instead of Necessity we used Possibility in (2), then every sequence
of fuzzy numbers would obey the law of large numbers.
Lemma 3.5 Let T be a t-norm and let ξ1 , ξ2 , . . . be a sequence of fuzzy numbers with modal values M1 ,
M2 , . . . then with the notations ηn = ξ1 + · · · + ξn , mn = (M1 + · · · + Mn )/n we have
Pos(
ηn
= mn ) = 1, n ∈ N
n
Proof. From Lemmas it follows that (ηn /n)(mn ) = 1, n ∈ N . Which ends the proof. The theorem in
question can be stated as follows:
Theorem 3.1 (Law of large numbers for fuzzy numbers) Let T ≤ H0 and let ξi = (Mi , α), i ∈ N be
fuzzy numbers. If
M1 + · · · + Mn
M = lim
n→∞
n
exists, then for any > 0,
ξ1 + · · · + ξn
lim Nes mn − ≤
≤ mn + = 1
n→∞
n
(5)
where
mn =
M1 + · · · + M n
.
n
Proof 3 If ≥ α then we get (5) trivially. Let < α, then from Lemma 1 and Lemma 2 it follows that
we need to prove (i) only for T = H0 . Using Lemma 4 we get
ηn
ηn Nes mn − ≤
≤ mn + = 1 − Pos
(−∞, mn − ) ∪ (mn + , ∞) =
n
n
−
sup
x∈[m
/ n −,mn +]
ηn
1 − |mn − (mn + )|/α
(x) = 1 −
=
n
1 + (n − 1)|mn − (mn ± )|/α
1−
1 − /α
1 + (n − 1)/α
and, consequently,
1 − /α
ηn
lim Nes mn − ≤
≤ mn + = 1 − lim
= 1.
n→∞ 1 + (n − 1)/α
n→∞
n
This ends the proof.
5
Remark 3.1 Theorem 1 can be interpreted as a law of large numbers for mutually T -related fuzzy variables [9]. Strong laws of large numbers for fuzzy random variables were proved in [7,8].
Remark 3.2 Especially, if T (u, v) = H1 (u, v) = uv then we get [5]
ξ1 + · · · + ξn
lim Nes mn − ≤
≤ mn + =
n→∞
n
n
ηn
= 1.
1 − (mn − ) = 1 − lim 1 −
n→∞
n
α
The following theorem shows that if T = ”min” then the sequence ξ1 = (Mi , α), ξ2 = (M2 , α) . . . does
not obey the law of large numbers for fuzzy numbers.
Theorem 3.2 Let T (u, v) = min{u, v} and ξi = (Mi , α), i ∈ N . Then for any positive , such that
< α we have
ξ1 + · · · + ξn
lim Nes mn − ≤
≤ mn + = .
n→∞
n
α
Proof 4 The proof of this theorem follows from the equalities ηn /n = (mn , α), n ∈ N and
ηn
= (M, α).
n→∞ n
lim
Figure 1: The limit distribution of ηn /n if T =”min”
Remark 3.3 From the addition rule of LR-type fuzzy numbers via sup-min convolution [2] it follows
that Theorem 2 remains valid for any sequence ξ1 = (M1 , α)LL , ξ2 = (M2 , α)LL , . . . of LL-type fuzzy
numbers with continuous shape function L.
4
Question
Let T be a t-norm such that H0 < T < ”min” and let ξ1 = (M1 , α), ξ2 = (M2 , α) . . . be a sequence of
symmetric triangular fuzzy numbers. Does this sequence obey the law of large numbers, i.e. does
lim ηn /n = χM
n→∞
follow from limn→∞ mn = M ?
5
Guess
If T is an Archimedean t-norm [10], then every sequence of fuzzy numbers ξ1 , ξ2 , . . . such that the
diameter (diam) of their supports are bounded by the same constant
diam(suppξ1 ) ≤ C, diam(suppξ2 ) ≤ C, . . . , diam(suppξn ) ≤ C,
obeys the law of large numbers for fuzzy numbers.
6
References
[1] P. L. Chebyshev, On mean quantities, Mat. Sbornik, 2(1867); Complete Works, 2(1948).
[2] D. Dubois and H.Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New
York, 1980.
[3] D.Dubois and H.Prade, Additions of Interactive Fuzzy Numbers, IEEE Transactions on Automatic
Control, 1981, Vol.AC-26, No.4 926-936.
[4] D.Dubois and H.Prade, Linear programming with fuzzy data, in:J.C.Bezdek Ed.,Analysis of Fuzzy
Information, Vol. 3: Applications in Engineering and Science, CRC Press, Boca Raton, FL, 1987
241-261.
[5] R.Fullér, On product-sum of triangular fuzzy numbers, Fuzzy Sets and Systems, 41(1991) 83-87
[6] H. Hamacher, Über logische Aggregationen nicht binär explizierter Entscheidung-kriterien, Rita
G.Fischer Verlag, Frankfurt, 1978.
[7] R. Kruse, The Strong Law of Large Numbers for fuzzy Random Variables, Information Sciences,
28(1982) 233-241.
[8] M. Miyakoshi and M.Shimbo, A Strong Law of Large Numbers for Fuzzy Random Variables.
Fuzzy Sets and Systems, 12(1984) 133-142.
[9] M. B. Rao and A.Rashed, Some comments on fuzzy variables, Fuzzy Sets and Systems, 6(1981)
285-292.
[10] B. Schweizer and A.Sklar, Associative functions and abstracts semigroups, Publ. Math. Debrecen,
10(1963) 69-81.
[11] L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1(1978)
3-28.
[12] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, I,
Information Sciences, (8)1975 199-249, II, Information Sciences, (8)1975 301-357, III, Information Sciences, (9)1975 43-80.
6
Follow ups
The results of this paper have been extended and improved in the following papers:
in journals
A28-c37 Shuming Wang; Junzo Watada, Some properties of T -independent fuzzy variables, MATHEMATICAL AND COMPUTER MODELLING, 53(2011), issues 5-6, pp. 970-984. 2011
http://dx.doi.org/10.1016/j.mcm.2010.11.006
7
Fullér [A28] proved a law of large numbers for T -related symmetric triangular fuzzy
variables with common spread by using necessity measure, where T (u, v) ≤ uv/(u +
v − uv) for all 0 ≤ u, v ≤ 1. Triesch [28] extended the results of Fullér [A28], a
nd studied some laws of large numbers for sequences of mutually T -related LR fuzzy
numbers, respectively, where T belongs to a class of continuous Archimedean t-norms.
Also utilizing necessity measure and continuous Archimedean t-norms, Hong and Ro
[29] further generalized the results of [28] to fuzzy numbers with unbounded supports.
Developing the ideas of [29], this paper further discusses the limit theorems for the sum
of T -independent fuzzy variables. (page 971)
A28-c36 Dug Hun Hong, Blackwell’s Theorem for T-related fuzzy variables, INFORMATION SCIENCES 180:(2010), issue 9, pp. 1769-1777. 2010
http://dx.doi.org/10.1016/j.ins.2010.01.006
Many different types of the law of large numbers for T-related fuzzy variables have
been studied by a number of authors, such as, Badard [1], Fullér [A28], Triesch [17],
Markov [13], Hong [4], Hong and Lee [6], Hong and Ro [5], and Hong and Ahn [7].
(page 1769)
A28-c35 Yanju Chen, Yan-Kui Liu A strong law of large numbers in credibility theory, WORLD JOURNAL OF MODELLING AND SIMULATION, Vol. 2 (2006) No. 5, pp. 331-337. 2006
http://www.worldacademicunion.com/journal/1746-7233WJMS/wjmsvol2no5paper06.pdf
Fullér [A28] studied a law of large numbers for symmetric triangular fuzzy variables
when T (u.v) ≤ uv/(u + v − uv). (page 331)
A28-c34 Hong DH, Renewal process with T-related fuzzy inter-arrival times and fuzzy rewards, INFORMATION SCIENCES, 176 (16): 2386-2395 AUG 22 2006
http://dx.doi.org/10.1016/j.ins.2005.06.008
Following Fullér (see [A28]), we say that ξ1 , ξ2 , . . . , ξn , . . . , obey the law of large
numbers if for all > 0 the quantity Nes(mn − < (ξ1 + ξ2 + · · · + ξn )/n < mn + )
tends to 1 as n → ∞. (page 2388)
A28-c33 Dug Hun Hong and Chul H. Ahn, Equivalent conditions for laws of large numbers for T-related
L-R fuzzy numbers, FUZZY SETS AND SYSTEMS, 136(2003) 387-395. 2003
http://dx.doi.org/10.1016/S0165-0114(02)00217-8
Abstract. In this paper, we give some necessary and sufficient conditions for laws
of large numbers for sequence of mutually T-related L-R fuzzy numbers when T is
an continuous Archimedean t-norm and diameter of support of the fuzzy numbers is
not uniformly bounded. We also consider some necessary and sufficient conditions for
laws of large numbers for L-R fuzzy random numbers, and generalize earlier results of
Fullér (Fuzzy Sets and Systems 45 (1992) 299-303), Triesch (Fuzzy Sets and Systems
58 (1993) 339-342) and Hong and Lee (Fuzzy Sets and Systems 121 (2001) 537-543).
..
.
In 1982, Badard [1] proved a law of large numbers for fuzzy numbers with common
spread when T (u, v) = uv. In 1991, Williamson [27] generalized law of large numbers
8
for fuzzy numbers, but his result was shown to be incorrect by Fullér and Triesch [A22].
In 1992, Fullér [A28] proved a law of large numbers for sequence of mutually T-related
symmetric triangular fuzzy numbers with common spreads if T (u, v) ≤ H0 (u, v) :=
uv/(u + v − uv) for all 0 ≤ u, v ≤ 1. (page 387)
Strong laws of large numbers for sums of independent fuzzy random variables have
been studied by several people. A SLLN for sums of independent and identically distributed (i.i.d.) fuzzy random variables was obtained by Kruse [18], and a SLLN for
sums of independent fuzzy random variables by Miyakoshi and Shimbo [21]. Also,
Klement et al. [ 17] proved some limit theorems which includes a SLLN, and Inoue
[16] obtained a SLLN for sums of independent tight fuzzy random sets, Hong and Kim
[12] studied Marcinkiewicz-type law of large numbers for fuzzy random variables under additional assumption. On the other hand, Näther et al. [22] considered a linear
regression model when centers and spreads are random variables. In this paper, we
also consider laws of large numbers for T-related L-R fuzzy numbers when centers and
spreads are i.i.d. random variables and generalize the result of Fullér [A28]. (page 388)
A28-c32 Dug Hun Hong, T-sum of L-R fuzzy numbers with unbounded supports, COMMUNICATIONS OF THE KOREAN MATHEMATICAL SOCIETY, 18 (2003), no. 2, 385-392. 2003
A28-c31 Dug Hun Hong and Jaejin Lee A convergence of geometric mean for T-related fuzzy numbers
FUZZY SETS AND SYSTEMS, 121(2001) 537-543. 2001
http://dx.doi.org/10.1016/S0165-0114(99)00136-0
Fullér [A28] studied a convergence of arithmetic mean for sequences of mutually Trelated symmetrict riangular fuzzynumbers with common spread if T (u, v) ≤ H0 (u, v) :=
uv/(u + v − uv) for all 0 ≤ u, v ≤ 1. Triesch [11] and Fullér [C47] generalize Fullér’s
results for certain sequences of L-R fuzzy numbers with uniformly bounded spreads if
T belongs to the class of Archimedian t-norms. Hong and Kim [9] generalized Triesch’s
results for sequences of fuzzy numbers in a Banach space and Hong [7] provided a general law of large numbers fo rarrays of L-R fuzzy numbers. Recently, Hong and Hwang
[8] provided new results regarding the effective practical computation of t-norm-based
products of R-type fuzzy numbers and their limit. Then it is natural to ask about convergence of geometric mean for fuzzy numbers. (page 537)
Now we can also ask the same type of conjecture as in Fullér’s paper [A28]: Suppose we
are given an Archimedean t-norm and a sequence ã1 , ã2 , . . . , ãn , . . . of fuzzy numbers
such that for all n, the diameter of the support of ãn is bounded by the same constant
C independent of n. Then (2) in Theorem 1 holds. (page 542)
A28-c30 Hong DH, Lee J, On the law of large numbers for mutually T-related L-R fuzzy numbers,
FUZZY SETS AND SYSTEMS, 116 (2): 263-267 DEC 1 2000
http://dx.doi.org/10.1016/S0165-0114(98)00071-2
Recently, Fullér [A28] proved a law of large numbers for sequences of mutually Trelated symmetric triangular fuzzy numbers with common spread if T (u, v) ≤ H0 (u, v) :=
uv/(u + v − uv) for all 0 ≤ u, v ≤ 1. Triesch [10], Hong [5], Hong and Hwang [7] and
Fullér [4] generalize Fullér’s results for certain sequences of L-R fuzzy numbers with
uniformly bounded spreads if T belongs to the class of Archimedean t-norms. Hong
9
and Kim [8] generalized Triesch’s results for sequences of fuzzy numbers in a Banach
space and Hong [6] provided a general law of large numbers for arrays of L-R fuzzy
numbers. Triesch stated in concluding remarks at the end of [11] that there could be
laws of large numbers for interesting classes of t-norms and sequences of fuzzy numbers where the diameter of the support of fuzzy numbers is not uniformly bounded.
The object of this paper is to study a law of large numbers concern- ing the Triesch’s
statement and generalize results of Fullér [A28] and Triesch [11]. (page 263)
A28-c29 Hong DH, Ro PI, The law of large numbers for fuzzy numbers with unbounded supports,
FUZZY SETS AND SYSTEMS, 116 (2): 269-274 DEC 1 2000
http://dx.doi.org/10.1016/S0165-0114(98)00188-2
Recently, Fullér [A28], Triesch [11], Hong [6], Hong and Kim [8] studied laws of large
numbers for fuzzy numbers. (page 269)
A28-c28 Markova-Stupnanova A T-law of large numbers for fuzzy numbers, KYBERNETIKA, 36 (3):
379-388. 2000
http://www.kybernetika.cz/content/2000/3/379
Fuller [2] introduced the law of large numbers for special LR-fuzzy numbers and tnorms bounded by the Hamacher product t-norm.
Theorem 1. (The law of large numbers, [2]) (page 381)
A28-c27 L.C.Jang and J.S.Kwon, A note on law of large numbers for fuzzy numbers in a Banach space,
FUZZY SETS AND SYSTEMS, 98(1998) 77-81. 1998
http://dx.doi.org/10.1016/S0165-0114(96)00391-0
Fullér [A28] proved a law of large numbers for sequences of symmetric triangular fuzzy
numbers with common spread and Triesch [3] generalizes Fullér’s result for sequences
of L-R fuzzy numbers with bounded spreads. In this paper, we study a law of large
numbers for sequences of fuzzy numbers in a Banach space to generalize earlier results
mentioned above. The fuzzy numbers considered need not have (uniformly) bounded
supports. Our condition relates the additive generator of t-norm and shape functions of
fuzzy numbers involved. (page 77)
A28-c26 A. Marková-Stupňanová, The law of large numbers for fuzzy numbers, Bulletin for Studies
and Exchanges on Fuzziness and its Applications, 75(1998) 53-59. 1998
Fullér [A28] introduced the law of large numbers for special LR-fuzzy numbers and
Hamacher t-norm. (page 54)
A28-c25 A. Marková-Stupňanová, A note on the recent results on the law of large numbers for fuzzy
numbers, Bulletin for Studies and Exchanges on Fuzziness and its Applications, 76(1998) 12-18.
Fullér [A28] proved a law of large numbers for sequences of symmetric triangular fuzzy
numbers with common spread. (page 12)
10
A28-c24 D.H.Hong and S.Y.Hwang, The convergence of T-product of fuzzy numbers, FUZZY SETS
AND SYSTEMS, 85(1997) 373-378. 1997
http://dx.doi.org/10.1016/0165-0114(95)00333-9
Recently, t-norm-based addition of fuzzy numbers and its convergence have been studied [A32, A30, A28, A24, 6, 7, 9]. (page 373)
A28-c23 A.Markova, T-sum of L-R fuzzy numbers, FUZZY SETS AND SYSTEMS, 85(1997) 379384. 1997
http://dx.doi.org/10.1016/0165-0114(95)00370-3
(i) Fullér’s theorem [A28] on the law of large numbers is valid for each t-norm T with
additive generator f and L-R fuzzy numbers such that both f ◦ L and f ◦ R are convex.
Consequently, the same is true, for given shapes L, R, for each t-norm T ∗ ≤ T . Note
that Fullér’s result concerns Hamacher’s t-norm . . . (page 383)
A28-c22 D.H.Hong and C. Hwang, A T-sum bound of LR-fuzzy numbers, FUZZY SETS AND SYSTEMS, 91(1997) 239-252. 1997
http://dx.doi.org/10.1016/S0165-0114(97)00144-9
As applications, we can consider Fullér’s theorem [A28] on the law of large numbers.
His result was generalized by Triesch [18], Hong [7] and Hong and Kim [10]. But, in
these results the uniform boundedness of supports of LR-fuzzy numbers is assumed.
Triesch [19] stated that there could be laws of large numbers for interesting classes of
t-norm and sequences of fuzzy numbers where the diameter of the support of fuzzy
numbers is not uniformly bounded. Using the idea of Theorem 5, Hong and Lee [11]
have s tudied a law of large numbers without assuming uniform boundedness and generalized results of Fullér [A28] and Triesch [19]. (page 251)
A28-c21 K.-L. Zhang and K. Hirota, On fuzzy number lattice (R̃, ≤), FUZZY SETS AND SYSTEMS,
92(1997) 113-122. 1997
http://dx.doi.org/10.1016/S0165-0114(96)00164-9
A28-c20 D.H.Hong and Y.M.Kim, A law of large numbers for fuzzy numbers in a Banach space,
FUZZY SETS AND SYSTEMS, 77(1996) 349-354. 1996
http://dx.doi.org/10.1016/0165-0114(95)00048-8
Recently, Fullér [A28] proves a law of large numbers for sequences of symmetric triangular fuzzy numbers with common spread and Triesch [9] generalizes Fullér’s results
for sequences of L-R fuzzy numbers with bounded spreads. The object of this paper is
to consider fuzzy numbers in a Banach space and to generalize earlier results of Fullér
and Triesch. (page 349)
A28-c19 Hong DH A convergence theorem for arrays of L - R fuzzy numbers INFORM SCIENCES
88 (1-4): 169-175 JAN 1996
http://dx.doi.org/10.1016/0020-0255(95)00160-3
11
A28-c18 A.Markova, Addition of L-R fuzzy numbers, Bulletin for Studies and Exchanges on Fuzziness
and its Applications, 63(1995) 25-29. 1995
A28-c17 D.H.Hong, A note on the law of large numbers for fuzzy numbers, FUZZY SETS AND
SYSTEMS 64(1994), No. 1, 59-61. [MR 95e:03152]
http://dx.doi.org/10.1016/0165-0114(94)90006-X
Abstract: We prove Fullér’s open question [Fuzzy Sets and Systems, 45(1992) 299303]: We construct a t-norm T showing Fullér’s theorem on . . . (page 59).
A28-c16 D.H.Hong, A note on the law of large numbers for fuzzy numbers, FUZZY SETS AND
SYSTEMS, 68(1994), No. 2, 243-243. [MR 1 321 502]
http://dx.doi.org/10.1016/0165-0114(94)90050-7
A28-c15 E.Triesch, Characterisation of Archimedean t-norms and a law of large numbers, FUZZY
SETS AND SYSTEMS, 58(1993) 339–342. 1993
http://dx.doi.org/10.1016/0165-0114(93)90507-E
Abstract: We study a law of large numbers for mutually T -related fuzzy numbers where
T is an Archimedean t-norm and extend earlier results of Fullér in this area. In particular, we show that the class of Archimedean t-norms can be characterized by the validity
of a very general law of large numbers for sequences of L-R fuzzy numbers. numbers.
(page 339)
Following Fullér (see [A28]), we say that ξ1 , ξ2 , . . . , ξn , . . . , obeys the law of large
numbers if for all > 0 the quantity Nes(mn − < (ξ1 + ξ2 + · · · + ξn )/n <
mn + ) tends to 1 for n → ∞. Fullér proves a law of large numbers for sequences
of symmetric triangular fuzzy numbers with common spread of limn→∞ mn exists and
T (u.v) ≤ H0 (u, v) := uv/(u + v − uv) for all 0 ≤ u, v ≤ 1. In [C27], the result is
extended to certain sequences of L-R fuzzy numbers if T belongs to a subclass of the
class of Archimedean t-norms. Fullér proposes the following conjecture at the end of
[A28]: Suppose we are given an Archimedean t-norm and a sequence ξ1 , ξ2 , . . . , ξn ,
. . . such that, for all n, the diameter of the support of ξn is bounded by the same constant
C independent of n. Then ξ1 , ξ2 , . . . , ξn , . . . obeys the law of large numbers. (pages
339-340)
in proceedings and in edited volumes
A28-c12 Baogui Xin; Tong Chen; Wei Yu; Weak Laws of Large Numbers for fuzzy variables based
on credibility measure, Seventh International Conference on Fuzzy Systems and Knowledge Discovery (FSKD), 10-12 August 2010, Yantai, Shandong, China, [ISBN 978-1-4244-5931-5], pp.
377-381. 2010
http://dx.doi.org/10.1109/FSKD.2010.5569638
A lot of researches have been reported in the area of WLLN for fuzzy numbers based
on fuzzy sets theory such as Fuller [A28] , Triesch [8], Hong [9], Hong and Kim [10],
Jang and Kwon [11], Hong and Ro [12], Taylor, Seymour and Chen [13], and in the
12
area of WLLN for fuzzy random variables based on credibility measure such as Yang
and Liu [14], but all of them are essentially different from WLLN for fuzzy variables
based on credibility measure. (page 377)
A28-c11 Pedro Terán, On Convergence in Necessity and Its Laws of Large Numbers, in: Didier Dubois
et al eds., Soft Methods for Handling Variability and Imprecision, Advances in Soft Computing,
vol. 48/2010, Springer, [ISBN 978-3-540-85026-7], pp. 289-296. 2010
http://dx.doi.org/10.1007/978-3-540-85027-4_35
A28-c10 Zhang, Guo-Chun; Wang, Shu-Ming; Dai, Xiao-Dong, The Convergent Results about the Sum
of Fuzzy Variables and the Law of Large Numbers, International Conference on Machine Learning
and Cybernetics, 19-22 Aug. 2007, vol.2, pp.1209-1214. 2007
http://dx.doi.org/10.1109/ICMLC.2007.4370328
In credibility theory, fuzzy variable introduced by Kaufmann [10] is a critical concept. In recent years, a lot of work has been done on the properties about the sum
of fuzzy variables or fuzzy numbers, and various forms of law of large numbers for
fuzzy variables were proposed. Badard [1] studied the behavior of arithmetical mean
for T -independent fuzzy numbers where T =”min” and T =classical product; Fullér
[A28] proved a law of large numbers for T -independent fuzzy numbers with common
spread if T (x, y) ≤ xy/(x + y − xy) for all 0 ≤ x, y ≤ 1; Triesch [21], Hong and
Hwang [4] generalized Fullér’s results for certain sequence of L - R fuzzy numbers if
T is a Archimedean t-norm; Hong and Lee [5] studied the law of large numbers for
T -independent L - R fuzzy numbers where T is an Archimedean t-norm and the diameter of the support of the fuzzy numbers is not uniformly bounded, this generalized
Triesch’s results.
..
.
This paper studies the convergent properties on the sum of T-independent fuzzy variables based on credibility theory and the convex hull of a function, and finally gives a
strong law of large numbers for fuzzy variables. (page 1209)
A28-c9 YJ Chen, TG Fan, FF Hao, The convergence theorems for sequence of fuzzy variables, Proceedings of the 5th International Conference on Machine Learning and Cybernetics, Aug. 13-16,
2006, Vols 1-7, pp. 1889-1894. 2006
http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=04370350
A28-c8 Dick, S. and Kandel, A. Granular computing in neural networks, in: Granular Computing:
An Emerging Paradigm, W. Pedrycz, Ed. Physica-Verlag GmbH, Heidelberg, Germany, 275-305.
2001
A28-c7 M. Oussalah, New Formulations of Law of Large Numbers and Its Convergence in the Framework of Possibility Theory, In: Da Ruan ed., Intelligent Techniques and Soft Computing in Nuclear
Science and Engineering: Proceedings of the 4th International Flins Conference Bruges, Belgium,
World Scientific, [ISBN 9789810243562], pp. 80-86. 2000
A28-c6 D.Dubois, E.Kerre, R.Mesiar and H.Prade, Fuzzy interval analysis, in: Didier Dubois and Henri
Prade eds., Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, Volume 7, Kluwer
Academic Publishers, [ISBN 0-7923-7732-X], 2000, 483-581. 2000
13
A28-c5 E. Pap, The law of large numbers in representation on uncertainty, in: B.De Baets, J. Fodor
and L. T. Kóczy eds., Proceedings of the Fourth Meeting of the Euro Working Group on Fuzzy
Sets and Second International Conference on Soft and Intelligent Computing (Eurofuse-SIC’99),
Budapest, Hungary, 25-28 May 1999, Technical University of Budapest, [ISBN 963 7149 21X],
1999 459-464. 1999
Following Fullér [A28] we say that ã1 = (a1 , α), ã2 = (a2 , α), . . . obeys the law of
large numbers if
1
lim (ã1 ⊕T · · · ⊕T ãn )(z) = χa (z)
n→∞ n
where
a1 + · · · + an
a = lim
n→∞
n
(page 460)
Fullér proves a law of large numbers if T ≤ T0H , where T0H is the Hamacher norm with
parameter 0; and for TM (where TM is min-t-norm) the law of large numbers is not
valid. Triesch [12] generalized Fullér’s result for sequences of LR-fuzzy numbers with
bounded spreads. Hong and Kim [5] generalized earlier results of Fullér and Triesch
and gave law of large numbers in Banach space. Fullér suggests the question: Does the
law of large numbers valid for t-norm T such that T0H ≤ T ≤ TM ? (page 461)
A28-c4 D.H.Hong and C. Hwang, Upper bound of T-sum of LR-fuzzy numbers, in: Proceedings of
IPMU’96 Conference (July 1-5, 1996, Granada, Spain), 1996 343-346.
A28-c3 D.Dubois, H.Prade and R.R.Yager eds., Readings in Fuzzy Sets for Intelligent Systems, Morgan
Kaufmann Publisher, San Mateo, 1993, page 24.
Studies the limit of the arithmetic mean of identical fuzzy numbers under various tnorm based extension principles. (page 24)
in books
A28-c2 Jorma K. Mattila, Text Book of Fuzzy Logic, Art House, Helsinki, [ISBN 951-884-152-7], 1998.
A28-c1 R. Lowen, Fuzzy Set Theory, Kluwer Academic Publishers, [ISBN: 0-7923-4057-4], 1996.
14