On law of large numbers for L-R fuzzy
numbers∗
Robert Fullér †
rfuller@ra.abo.fi
Abstract
This paper extends the author’s earlier work on the Law of Large Numbers
for fuzzy numbers [2] to the case where the fuzzy numbers are of type L-R.
Namely, we shall define a class of Archimedean triangular norms in which the
equality
lim Nes(mn − ≤ ηn ≤ mn + = 1, for any > 0,
n→∞
holds for all sequences of fuzzy numbers,
ξi = (Mi , α, β)LR , i ∈ N,
with twice differentiable and concave shape functions L and R; where
ηn =
ξ1 ⊕ · · · ⊕ ξn
, n ∈ N,
n
is the sequence of t-arithmetic means of the first n-terms of {ξi } (defined via a
sup-t-norm convolution),
mn =
M 1 + · · · + Mn
,
n
and Nes denotes Necessity.
Keywords: Triangular norm, L-R fuzzy number, t-arithmetic mean, possibility, necessity
1
Definitions
A fuzzy number ξ is a fuzzy set of the real line IR with an unimodal, normalized (i.e.
there exists unique a ∈ IR such that ξ(a) = 1) and upper-semicontinuous membership
function. An L-R fuzzy number ξ denoted by (M, α, β)LR is defined as ξ(t) = L((M −
t)/α) if M − α ≤ t ≤ M , ξ(t) = R((t − M )/β) if M ≤ t ≤ M + β and ξ(t) = 0
∗
The final version of this paper appeared in: R.Lowen and M.Roubens eds., Proceedings of the
Fourth IFSA Congress, Vol. Mathematics, Brussels, 1991 74-77.
†
Supported by the German Academic Exchange Service (DAAD).
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otherwise, where M ∈ IR is the modal value and α > 0, β > 0 are the left and right
spreads of ξ, respectively.
Given a subset D ⊂ IR, the grade of possibility of the statement ”D contains the
value of ξ” is defined by [7]
P os(ξ|D) = sup ξ(x)
x∈D
The quantity 1 − P os(ξ|D̄), where D̄is the complement of D, is denoted by N es(ξ|D)
and is interpreted as the grade of necessity of the statement ”D contains the value of
ξ”.
If D = [a, b] ⊂ IRthen instead of N es(ξ|[a, b]) we shall write N es(a ≤ ξ ≤ b) and if
D = x, x ∈ IR we write N es(ξ = x). Let {ξn } 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), for all x ∈ IR.
n→∞
A triangular norm (t-norm for short) T is said to be Archimedean iff T is continuous
and T (x, x) < x, for all x ∈ (0, 1).
Every Archimedean t-norm T is representable by a continuous and decreasing function
f : [0, 1] × [0, ∞] with f (1) = 0 and
T (x, y) = f [−1] (f (x) + f (y))
where f [−1] is the pseudo-inverse of f , defined by f −1 (y) if y ∈ [0, f (0)] and f [−1] (y) = 0
if y ∈ [f (0), ∞]. The function f is the additive generator of T .
Let T be a t-norm and let ξ1 , ξ2 be fuzzy sets of the real line, then their T -sum ξ1 ⊕ ξ2
is defined by [1]
(ξ1 ⊕ ξ2 )(z) =
sup T (ξ1 (x1 ), ξ2 (x2 )), z ∈ IR
x1 +x2 =z
Let n ∈ N , T be a t-norm and ξ1 , . . . , ξn be fuzzy sets of IR. Then (according to
Zadeh’s extension principle) their T-arithmetic mean (ξ1 ⊕ · · · ⊕ ξn )/n is defined as
ξ1 ⊕ ... ⊕ ξn
(x) := (ξ1 ⊕ · · · ⊕ ξn )(nx), for all x ∈ IR.
n
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The result
The theorem in question can be stated as follows:
Theorem 2.1 Let T be an Archimedean t-norm with an additive generator f and let
{ξi = (Mi , α, β)LR , i ∈ N} be a sequence of fuzzy numbers of type L-R. If L and R are
twice differentiable, concave functions, and f is twice differentiable, strictly convex
function and
m := lim mn
n→∞
exists and it is finite, then
lim N es(mn − ≤ ηn ≤ mn + ) = 1,
n→∞
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for all > 0, where {ηn = (ξ1 ⊕ · · · ⊕ ξn )/n, n ∈ N} is the sequence of T -arithmetic
means of the first n-terms of {ξi } and
mn =
M1 + · · · + Mn
.
n
Proof. From Theorem 1 [3] it follows that

1 




[−1]

 f
nf L[(mn − z)/α]

ηn (z) =

[−1]

f
nf R[(z − mn )/β]





 0
if z = mn
if mn − α ≤ z ≤ mn
if mn ≤ z ≤ mn + β
otherwise
therefore, we have
lim N es(mn − ≤ ηn ≤ mn + ) =
n→∞
lim (1 − Pos(ηn | (−∞, mn − ) ∪ (mn + , ∞))) =
n→∞
1 − lim max{f [−1] (nf (L(/α))), f [−1] (nf (R(/β)))}
n→∞
= 1 − f [−1] ( lim n max{f (L(/α)), f (R(/β})
n→∞
finally, from f (L(/α)) > 0, f (R(/β)) > 0 and limx→∞ f [−1] (x) = 0 we get (i).
(ii) Let z ∈ IR be arbitrarily fixed. If z = m then we have
( lim ξn )(m) = lim ξn (mn ) = 1.
n→∞
n→∞
Without lost of generality we can suppose that z < m. Then there exist an index
n0 ∈ N and a real
number > 0 such that mn − z > for all n ≥ n0 Therefore we get
( lim ξn )(z) = lim ξn (z) = lim f [−1] (nf (L((mn − z)/α))) ≤
n→∞
n→∞
n→∞
f [−1] ( lim nf (L(/α))) = 0,
n→∞
which proves the theorem.
Remark 2.1 Theorem 1 can be interpreted as a Law of Large Numbers for mutually
T -related fuzzy variables [6]. Strong laws of large numbers for fuzzy random variables
were proved in [4,5].
Remark 2.2 If T (x, y) = min{x, y} (non-Archimedean t-norm) then for any 0 < <
min{α, β} we get
lim N es(mn − ≤ ηn ≤ mn + ) = 1 − max{L(/α), R(/β)}
n→∞
Nes( lim ηn = m) = 0
n→∞
which shows that Theorem 1 is not valid for the whole family of t-norms.
3
m-α
m
m+β
Figure 1: The limit distribution of ηn , ηn = (m, α, β)LR , if T =’min’.
References
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[2] R.Fullér, A Law of Large Numbers for Fuzzy Numbers, Fuzzy Sets and Systems,
45(1992) 299-303.
[3] R. Fullér and T. Keresztfalvi, t-Norm-based Addition of Fuzzy Intervals, Fuzzy
Sets and Systems 51(1992) 155-159.
[4] R.Kruse, The Strong Law of Large Numbers for fuzzy Random Variables, Information Sciences, 28(1982) 233-241.
[5] M. Miyakoshi and M. Shimbo, A Strong Law of Large Numbers for Fuzzy Random Variables, Fuzzy Sets and Systems, 12(1984) 133-142.
[6] M.B. Rao and A. Rashed, Some Comments on Fuzzy Variables, Fuzzy Sets and
Systems, 6(1981) 285-292.
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