A note on the law of large numbers for fuzzy variables ∗
Robert Fullér
rfuller@abo.fi
Eberhard Triesch
triesch@math2.rwth-aachen.de
Abstract
This short note a counterexample showing that Williamson’s theorem on the law of large numbers
for fuzzy variables under a general triangular norm extension principle is not valid.
The objective of this note is to provide a counterexample to Theorem 1 in Williamson’s paper [1]. To
save space, we essentially use the same notation as in [1] and we do not repeat the statement of the
theorem. Let t(u, v) := uv (product norm) and define the sequence of fuzzy numbers (Xi )∞
i=1 by their
membership functions as follows:
1 − |x|i , if −1 ≤ x ≤ 1,
µXi (x) :=
0,
otherwise
Then αXi = βXi = 0P
for all i and Theorem 1 states that the membership functions µZN of the arithmetic
means ZN = (1/N ) N
i=1 Xi converge pointwise (as N → ∞) to the function µ given by
1, for x = 0,
µ(x) :=
0, otherwise
at least on (−1, 0) ∩ (0, 1). However, we will show that in the open interval (-1, 1), the functions µZN
are bounded from below by some strictly positive function (not depending on N). To see this, recall that
the T -arithmetic means µZN are defined by
µZN (z) =
N
Y
sup
x1 +...xN =Nz i=1
µXi (xi ).
For z ∈ (−1, 1), we can thus estimate
µZN (z) ≥
N
Y
µXi (z) =
i=1
N
∞
Y
Y
(1 − |z|i ) ≥
(1 − |z|i ).
i=1
i=1
By Euler’s pentagonal number theorem (see, e.g., [2], p.312), the infinite product is well known to
converge for |z| < 1 to the following series:
f (|z|) = 1 +
∞
X
2 −n)/2
(−1)n (|z|(3n
2 +n)/2
+ |z|(3n
).
n=1
The value of the infinite product is thus positive on the interval (−1, 1).
∗
The final version of this paper appeared in: R. Fullér and E. Triesch, A note on the law of large numbers for fuzzy variables,
Fuzzy Sets and Systems, 55(1993) 235-236. doi: 10.1016/0165-0114(93)90136-6
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References
[1] R.C.Williamson, The law of large numbers for fuzzy variables under a general triangular norm
extension principle, Fuzzy Sets and Systems, 41(1991) 55-81.
[2] T.M.Apostol, Introduction to Analytic Number Theory, (Springer Verlag, Berlin-HeilderbergNew York, 1976).
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