The strong law of large numbers for fuzzy random variables

INFORMATION

SCIENCES

X&233-241

(1982)

233

The Strong Law of Large Numbers for Fuzzy Random Variables RUDOLF KRUSE

Institut f&r Angewandte Mathe~t~k der Tech~is~he~Uni~~r~~t~t~rau~~hwei~ Braunschweig, West Gemany

ABSTRACT

Sequences of independent and identicaily distributed fuzzy random variables are considered. It is shown that the strong law of large numbers holds also for fuzzy random variables. This result is used to give a consistent estimator for the expectation of a fuzzy random variabie.

1. INTRODUCTION Fuzzy random variables are random variables whose values are not real but fuzzy numbers. A fuzzy number may assume different real values, with each of which a degree of acceptability is associated. Kwakernaak [l] characterized a fuzzy random variable as a particular kind of fuzzy set. The expectation of a fuzzy random variable X is then defined as the image of the fuzzy set representing X under an appropriate mapping. In this paper we consider sequences of independent and identically distributed fuzzy random variables. It is shown that the strong law of large numbers holds also for fuzzy random variables. Section 2 of the paper describes briefly some properties of fuzzy numbers (see also M. Mizumoto and T. Tanaka [3]). In Section 3 we introduce formally the notions of fuzzy random variables and expectation of fuzzy random variables. For the motivation of these notions we refer to Kwakernaak [ 1, 21. In Section 4 the law of large numbers is proved and two examples are considered. Finally a consistent estimator for the expectation of a fuzzy random variable is given. Other definitions of fuzzy random variables and expectations have been given by Zadeh [4] and Nguyen [S] (in connection with linguistic variables), Nahmias [6, 71, Stein and Talati [8], and Hirota [9].

OElsevier Science Publishing Co., Inc. 1982 52 Vanderbilt Ave., New York, NY 10017

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RUDOLF KRUSE

234

2.

FUZZY RUMBAS

A fuzzy number (see Dubois and Prade [lo]) f in the real Line R is a fuzzy set f: R --* [O, I] that satisfies the following properties: f is piecewise continuous,

(2.1)

there exists an element x E R such that f ( x ) = 1,

(2.2)

and f is convex, i.e. if

x,,x~E!?

and X=[O,l],

then

f(hx,+(l-h)x,)~f(x,)Af(x,).

(2.3)

Let F(R) denote the set of all fuzzy numbers. A useful tool for the treatment of fuzzy numbers is the level sets considered by Negoita [I I]. The $evel-set Nf(&) of a fuzzy number f is the nonfuzzy set defined by o
N,(p) : = {x E Rlf (xl 2 ~1,

(2.4)

A fuzzy number may be decomposed into its level sets by

where Q is the set of the rational numbers. Let f and g be two fuzzy numbers in R. Then (by the extension principle: Zadeh [4]) two operations for fuzzy numbers are defined by

tf+

g)(a) :==

sup

{f (X)h g(v)>

sup

Cf(x)A dYN.

r,yeR:x+y=a

(f * g)(a):==

x,ysR:x*y-a

(2.6)

Consequently, if p is a real positive constant, then (p*f)(a)=

sup I,yE.:x*y_-o{f(X)h~~~ftY)J=f(u/~)‘

(2.7)

235

STRONG LAW OF LARGE NUMBERS 3.

FUZZY RANDOM VARIABLES

Let (51,5, P) be a probability space. A fuzzy random variable X is defined as a map X: D + F(R), specified by

that satisfies the following properties: (1) For each 111 E (0, I] both Uz and Uz* defined by

(3.1)

t.$+*(0) := supiv&) are finite real valued random variables on (Sz,9, P) such that the mathematical expectations EUZ and EU:' exist. (2) For each o E Q and each ~1E (0, I]

Let X be a fuzzy random variable on (52,9, P). Denote by a(X) the sigma algebra of subsets of fz generated by the random variables U:, y E (0, I], and U;L*, p f(02I]. F~e~ore let (SJ’,%‘, P’) be an nonatomic probabi~ty space, and let (&?,5, P) = (Q x W, 9%~Q’, P CO ?,I, where PC+P'is the product measure on the product sigma algebra 9@ 9’. Then the set % of all possible originals of X is defined as the set of all random variables defined on (& 5, P) that are o(X)@%‘-measurable. Kwakemaak [I] used this concept to give a proper definition for the notion of the expectation of a fuzzy random variable. DEFINITION1. The expectation of a fuzzy random variable X is the fuzzy number EX defined by

(EX)(a):=

inf X$Ji(@, w’)), -sup _ oeo tiE5CEU-=a CJE,Q

UER.

(3.3)

If X, and X, are two fuzzy random variables on (a, 9, P),then the sum X, + X, and the product Xi * X, are fuzzy random variables on the probability space (bz, 9, P) defined by

(4 -t x2)&?:=ML +(x2),9 (X1 * X2)&%:=(X*1, ‘(X,),.

(3.4)

RUDOLF

236 4.

THE LAW OF LARGE

KRUSE

NUMBERS

Let(X),i=1,2,3 ,..., be a sequence of fuzzy random variables. (Xi) is called a sequence of independent, identically distributed (iid) fuzzy random variables if the following holds: For each p E (0, l] both ( LJ$) and (u;tt)

defined by

(4.1)

are sequences of independent, identically distributed (usual) random variables. If the sigma algebras a(X,) of subsets of SZgenerated by Xi are independent, then (Xi) is a sequence of independent fuzzy variables. THEOREM 1. If (Xi) is a sequence of independent, identically distributed fuzzy random variables on (D, 9, P) such that

if infN,,,(p)

c~,cL’EQ~(O,~I < infNEX,(p’)

and p < cc’, then

and sup %x, ( CL ) ’ sup &x, ( p’) 7

(4.2)

then P-almost everywhere.

Proof. Let p E (0, l] be an arbitrary fori=1,2,3 ,..., andoE

It follows

real number.

(4.3)

By (3.1) and (3.2) we have

231

STRONG LAW OF LARGE NUMBERS Definition 1 gives for i = 1,2,3,. . .

From Kolmogoroff’s strong law of large numbers we know that there exists a set S,E~suchthatP(S,)=OandforwE~-SS, (4.4)

(4.5) and for We define S:= u peQn(O,,l,S,,. We have P(S)=O. For weti-S p E Q n(O, l] the equations (4.4), (4.5) are valid. To prove the theorem we have to show: If w E fz - S and x E W, then

= lim n-rm

sup pc (0.1 ]flQ

{P.Il(l/“)~:_,(i~,i(0). (l/nrGL,U;,?w)l

(4)

(4.6) We have to consider four cases. Let o E 6?- S and x E R. (1) (EX,)(x) = 1. Let p E (O,I)nQ. Then we have Et.& -c n < EU,Tf, and there exists E>O such that EU:,+E
238

RUDOLF KRUSE

and it follows that

Therefore

(2) 0 < (EX,)(x) < 1 and x < EU;,. (EX,)(x) -Cp2. Then we have

There exists E> 0 such that

We can obtain n Esuch that for all n > n e

ft follows for all n 3 n,

and

Therefore

Let p,, p2 E (0,l)na

such that p, -C

STRONG

LAW OF LARGE

239

NUMBERS

It follows that

For the cases (3) 0 < (EX,)(x) (4) (EX,)(x)

< 1, x > EUT;, and

= 0,

the conclusion

.Frn i

,$(X,)m)(x) =(EX,)(x) r-l

n

is proved in the same way.

REMARK 1. (4.2) is (for example) satisfied if, for all o E P and i = 1,2,3,. . . , is continuous and strictly increasing (decreasing) in ( - cc,inf Ncx,,,( l)]

(Xi),

([sup +,,,(l)Y

+ 00)).

EXAMPLE 1. Consider the class Z:= {f,,s]r ER, (Nahmias [6]) fuzzy numbers defined by

s E W’}

of all normal

We have f,,, +fr.,s,=fr+r,,s+sj and c* f,,, = f,,,,,. Let X be a fuzzy random variable on (Q, 9, P) such that if o E &2, then X, E Z. Consider the random variable (R,S) on (Q,T,P) defined by (R,S)(o)= (T,s) if and only if X, = f,,,. By Theorem 1 we have

EX=fm,,s. EXAMPLE 2. Consider an opinion poll, during which a number of individuals are questioned. The responses are classified into k categories respectively characterized by the fuzzy numbers f,, . . . ,fk. Randomness occurs because it is not known which response may be expected from any given individual. Suppose that the probabilities of fractions of respondents are unknown. We are interested in the expected fuzzy number. One method is the following: We ask n “with replacement”). Suppose the persons on their opinion (independently,

240

RUDOLF KRWSE

T,:==(ww”

The following thewem

is a ~~~e~~~

I*

F(R),

ofTkieorem I*

THEOREM 2. Let (Xi) be a sequence of (iid) fuz.zy random variables on (n, %, P) such that (4.2) is suti@‘@dfor all P E 8. Then

REFERENCES

2. N. Kwalcemaak, Puny random vtiables. Part 2: Algorithms and exaslples for the discrete case, hform. Sei. 1”?:25?-278(1979). 3. M. Mizumataand K. Tan&~ Soxne properties of fuzzy nnnxbers, in Advances h Fuzzy Set Theory a& ~~~1~~~~~~ (FG. M. Cup&, R. K. Ragade, and R. R. Yagq Fds.), NorthHolland, Amsterdam, 1979, pp. I.%- 165. 4. L. A. Zadeh, The concept of a Linguistic variable and its application in approximate reaoning, Part 2, Infurm. Sci. 8:301-357 (1975). 5. ES.T. Nguyen, On fuzziness and linguistic probabilities, J. Math. And. &pi. 61:658-671 (1977f.

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241

6. S. Nahmias, Fuzzy variables, Fuzzy Sets und Sysrem I :97-I 10 (1978). 7. S. Nahmias, Fuzzy variables in a random environment, in Advances in Fuzzy Set Theov and Applications (M. M. Gupta, R. K. Ragade, and R. R.‘Yager, Eds.), North-Holland, Amsterdam, 1979, pp. 165-180. 8. W. E. Stein and K. Talati, Convex fuzzy random variables, FUZZJ.JSets and Sysfems 6:217-283 (1981). 9. K. Hirota, Concepts of probabi~stic set, Fuzzy Sets and Systems $3 l-46 (198 1). 10. D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic, New York, 1980. 11. C. V. Negoita and D. A. Ralescu, Applications of Fuzzy Sets to System Analysis, Birkbiuser, Basel, 1975. 12. V, K. Rohatgi, An Intr~~ction to Fro&bili~ Theory and ~uthe~ticaf Sf~ristics, Wiley, New York, 1976. Received February 1982