elliptic boundary value problems with λ dependent boundary conditions abstract in this paper second order elliptic boundary value problems on bounded domains r n with boundary conditions on depending nonlinearly on the spectral parameter are investigated in an operator theoretic framework for a general class of locally meromorphic functions in the boundary condition a solution operator of the boundary value problem is constructed with the help of a linearization procedure in the special case of rational nevanlinna or riesz herglotz functions on the boundary the solution operator is obtained in an explicit form in the product hilbert space l 2 l 2 m which is a natural generalization of known results on linear elliptic boundary value problems and rational boundary value problems for ordinary second order differential equations

analysis of dynamical systems whose inputs are fuzzy stochastic processes this paper considers systems whose input signals are fuzzy stochastic processes of second order the analysis is entirely restricted to discrete time linear time invariant systems convergence conditions of the output are given the equations on the mean value functions and the covariance functions are derived the representation of fuzzy stochastic processes is also discussed

on the negative squares of indefinite sturm liouville operators abstract the number of negative squares of all self adjoint extensions of a simple symmetric operator of defect one with finitely many negative squares in a krein space is characterized in terms of the behaviour of an abstract titchmarsh weyl function near 0 and these results are applied to a large class of symmetric and self adjoint indefinite sturm liouville operators with indefinite weight functions

cones and decomposition of sub and supermartingales abstract we study sub and supermartingales with values in the class f c of all convex fuzzy subsets of a banach space having compact support the terms sub and super are understood with respect to a partial order not necessarily inclusion in this general framework orders with respect to which a doob type decomposition can hold are naturally associated to convex subcones with a special property termed f c properness decomposition theorems into the sum or difference of a martingale and a compensator are proven in the sub case the martingale need exist only in a superspace of f c

a multiobjective fuzzy stopping in a stochastic and fuzzy environment abstract in a stochastic and fuzzy environment a multiobjective fuzzy stopping problem is discussed the randomness and fuzziness are evaluated by probabilistic expectations and linear ranking functions respectively pareto optimal fuzzy stopping times are given under the assumption of regularity for stopping rules by using optimal stopping times

fuzzy set valued gaussian processes and brownian motions gaussian processes and brownian motion are concepts and tools in modelling important uncertain systems in many areas in view of uncertainty complexity in many real world problems we extend these tools to the case where stochastic processes can take on fuzzy sets as values in this paper we discuss fuzzy set valued gaussian processes based on the results of s li y ogura v kreinovich limit theorems and applications of set valued and fuzzy set valued random variables kluwer academic publishers dordrecht 2002 s li y ogura h t nguyen gaussian processes and martingales for fuzzy valued variables with continuous parameter inform sci 133 2001 7 21 s li y ogura f n proske m l puri central limit theorems for generalized set valued random variables j math anal appl 285 2003 250 263 n n lyashenko on limit theorems for sums of independent compact random subsets in the euclidean space j soviet math 20 1982 2187 2196 and m l puri d a ralescu the concept of normality for fuzzy random variables ann probab 13 1985 1373 1379 we also introduce the concept of fuzzy set valued brownian motion and then prove several properties of such processes

a pocket guide to nonlinear differential equations in musielak orlicz spaces abstract the musielak orlicz setting unifies variable exponent orlicz weighted sobolev and double phase spaces they inherit technical difficulties resulting from general growth and inhomogeneity in this survey we present an overview of developments of the theory of existence of pdes in the setting including reflexive and non reflexive cases as well as isotropic and anisotropic ones particular attention is paid to problems with data below natural duality in absence of lavrentiev s phenomenon

convergence theorem for fuzzy martingales abstract we study fuzzy set valued measures in a banach space and their relationships to fuzzy random variables our main result is a convergence theorem for fuzzy martingales our tool is a radon nikodym type theorem for fuzzy measures which are absolutely continuous with respect to probability measures

fuzzy probability measures abstract in 4 hohle has defined fuzzy measures on g fuzzy sets 2 where g stands for a regular boolean algebra consequently since the unit interval is not complemented fuzzy sets in the sense of zadeh 8 do not fit in this framework in a straightforward manner it is the purpose of this paper to continue the work started in 5 which deals with 0 1 fuzzy sets and to give a natural definition of a fuzzy probability measure on a fuzzy measurable space 5 we give necessary and sufficient conditions for such a measure to be a classical integral as in 9 in the case the space is generated a counterexample in the general case is also presented finally it is shown that a fuzzy probability measure is always an integral if the space is generated if we replace the operations and by the t norm to and its dual s0 see 6

on fuzzy measure and fuzzy integral on fuzzy set abstract on a fuzzy algebra of fuzzy sets we discuss some structural characteristics of a fuzzy measure the pseudo autocontinuity and some other concepts for a fuzzy measure are introduced and the relations between these cre discussed for the case of fuzzy sets riesz s theorems and lebesgue s theorems for a sequence of measurable functions are proved the concept of the fuzzy integral on a fuzzy set is given and some properties of fuzzy integrals on fuzzy sets are studied finally we prove some convergence theorems of a sequence of fuzzy integrals on fuzzy sets

fuzzy valued probability in this paper the main result is introduction of fuzzy valued probability restricted with respect to probability selectors in order to treat a highly imprecise probability system the fuzzy valued probability is defined over the measurable space as a fuzzy valued set function which is not additive in classical sense but it is additive with respect to restricted arithmetics some properties and the connection with fuzzy valued measure are discussed also an application in markov decision process with imprecise information is presented

mean square integral and differential of fuzzy stochastic processes abstract this paper defines the concepts of integral and differential in mean square associated with a class of fuzzy stochastic processes and studies their integrability and differentiability properties the definition and the mean square calculus properties of a gaussian fuzzy stochastic process are also discussed

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

parameter estimations of normal fuzzy variables abstract although fairly much attention has been paid to theoretical applications of nahmias fuzzy variables little effort has been focused on the related parameter estimation issues this keeps down the practical use of fuzzy variables in this note we make a preliminary discussion on parameter estimation of normal fuzzy variables with parameters a b and distinguish three situations estimation of a with b known estimation of b with a known and estimation of unknown a and b formulae for calculating parameter estimates by use of practical collected data are presented

separability for graph convergence of sequences of fuzzy valued random variables abstract we prove that the graph convergence in hausdorff metric or kuratowski mosco topology of a sequence of fuzzy sets follows from the convergence of the sequences of the level sets for countable dense levels as an application we give a strong law of large numbers for fuzzy valued random variables including the case when the level sets may not be bounded

on modeling possibilistic uncertainty in two state reliability theory in this paper we show how a possibilistic uncertainty model can be used to represent and manipulate uncertainty about the states of a system and of its components at the same time we present a thorough study of the incorporation of this possibilistic uncertainty model in classical two state reliability theory the possibilistic reliability of a component or system is introduced and studied furthermore we introduce the important notion of a possibilistic structure function based upon the concept of a classical two valued structure function under certain conditions of possibilistic independence it allows us to calculate the possibilistic reliability of a system in terms of the possibilistic reliabilities of its components finally we give straightforward methods for determining a possibilistic structure function from its classical two valued counterpart in this way we intend to show that a possibilistic uncertainty model in two state reliability theory is formally analogous to and certainly not more complicated than a probabilistic uncertainty model

strong law of large numbers with respect to a set valued probability measure in this paper we define the expected value of a random vector with respect to a set valued probability measure the concepts of independent and identically distributed random vectors are appropriately defined and a strong law of large numbers is derived in this setting finally an example of a set valued probability useful in bayesian inference is provided

elliptic eigenvalue problems with eigenparameter dependent boundary conditions abstract let a be a uniformly elliptic second order linear operator on a smooth bounded domain r n we study the eigenvalue problem au u subject to boundary conditions b0u b1u on where bj are linear boundary operators the problem is recast in the form a u u in a hilbert or krein space and results are given on the location and type of the spectrum full and half range completeness and regularity of critical points

on functionals representable by fuzzy measures necessary and sufficient conditions for a functional t to be representable by fuzzy measure are given the calculus of the minimal t and the maximal t fuzzy measures which represent the functional t and the fuzzy integrals f d t and f d t are made explicit in terms of t moreover it is shown that in case t is defined on a fuzzy domain of integration b every f b is rt measurable finally some functionals related to fuzzy additive measures are presented

on the convergence of fuzzy martingales the purpose of this paper is to prove the convergence theorem for fuzzy martingales without assuming that their values are lipschitz or continuous fuzzy numbers on rn this approach allows us to deduce many results on convergence of fuzzy numbers and fuzzy random variables from the relevant results on real numbers and real valued random variables that appear as their support functions based on the concept of fuzzy downward martingale and applications of convergence theorem of fuzzy martingales the strong law of large numbers for fuzzy random variables is proved here the convergence is in the uniform metric but not the separable metric on fuzzy number space

fuzzy measure of fuzzy events defined by fuzzy integrals abstract we define the concept of fuzzy measure of a fuzzy event by using a general form of fuzzy integral proposed by murofushi called fuzzy t conorm integral encompassing previous definitions zadeh defined the probability measure of a fuzzy event and later the possibility measure of fuzzy event using a duality property of fuzzy t conorm integral we propose a general definition of fuzzy measure of fuzzy events which is compatible with previous definitions of zadeh and possesses all properties of a fuzzy measure in particular the duality property using our definition we examine the case of decomposable measures and belief functions a comparison with previous works is provided

sturm liouville operators with indefinite weight functions and eigenvalue depending boundary conditions we consider a class of boundary value problems for sturm liouville operators with indenite weight functions the spectral parameter appears nonlinearly in the boundary condition in the form of a function which has the property that 7 is a generalized nevanlinna function we construct linearizations of these boundary value problems and study their spectral properties

existence and regularity for mixtures of micromagnetic materials a new model for the energy of a mixture of micromagnetic materials is introduced within the context of functions with special bounded variation existence and regularity for the solution of an optimal design problem in micromagnetics are obtained

on finite rank perturbations of definitizable operators it was shown by p jonas and h langer that a selfadjoint definitizable operator a in a krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator b is selfadjoint and the resolvent set b is nonempty it is the aim of this note to prove a more general variant of this perturbation result where the assumption on b is dropped as an application a class of singular ordinary differential operators with indefinite weight functions is studied

characterization of energy minimizers in micromagnetics we present a new characterization of minimizing sequences and possible minimizers all called the minimizing magnetizations for a nonlocal micromagnetic like energy without the exchange energy our method is to replace the nonlocal energy functional and its relaxation with certain local integral functionals on divergence free fields obtained by a two step minimization of some auxiliary augmented functionals through this procedure the minimization problem becomes equivalent to the minimization of a new local variational functional called the dual variational functional which has a unique minimizer we then precisely characterize the minimizing magnetizations of original nonlocal functionals in terms of the unique minimizer of the dual variational functional finally we give some remarks and ideas on solving the dual minimization problem

decomposition and representation of fuzzy valued measure abstract fuzzy valued measure is a mapping which maps a algebra to a set of fuzzy sets with convex compact level defined on a banach space x decomposition and representation of fuzzy valued measure by a sequence of single valued measures is given the case when the set of images is the set of fuzzy numbers that is when x r is considered

fuzzy valued measure abstract the main purpose of this paper is to introduce and develop the notion of a fuzzy valued measure in separable banach space this definition of fuzzy valued measure is a natural generalization of the set valued measure radon nikod m theorems for fuzzy valued measure are established

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

a krein space approach to symmetric ordinary differential operators with an indefinite weight function on considere les proprietes spectrales du probleme differentiel l f 1 n p 0 f n n 1 n 1 p 1 f n 1 n 1 p n f rf sur un intervalle fini ou infini a b avec des coefficients reels localement sommables 1 p 0 p 1 p n r sous l hypothese que p 0 0 et que la fonction poids r change de signe sur a b

decomposition theorems for fuzzy supermartingales and submartingales doob decomposition and riesz decomposition theorems in standard martingale theory are generalized to fuzzy martingales the concepts of two types of riesz decomposition for fuzzy supermartingales are given and the necessary and sufficient conditions of that fuzzy supermartingale or submartingale resp supermartingale has doob decomposition resp riesz decomposition are discussed in detail

sharp eigenvalue estimates for rank one perturbations of nonnegative operators in krein spaces abstract let a and b be selfadjoint operators in a krein space and assume that the resolvent difference of a and b is of rank one in the case that a is nonnegative and i is an open interval such that a i consists of isolated eigenvalues we prove sharp estimates on the number and multiplicities of eigenvalues of b in i the general result is illustrated with eigenvalue estimates for singular indefinite sturm liouville problems

generalized resolvents and the boundary value problems for hermitian operators with gaps a hermitian operator a with gaps j j 1 j m is studied the self adjoint extensions which put exactly kj eigenvalues into each gap j j in particular for kj 0 1 j m the extensions preserving the gaps are described in terms of boundary conditions the generalized resolvents of the extensions with the indicated properties are described also a solvability criterion and description of all the solutions of the hamburger moment problem with supports in r j 1m j j are obtained in terms of the nevanlinna matrix

optimal stopping problems in a stochastic and fuzzy system in a stochastic and fuzzy environment two kinds of stopping models are discussed and compared the optimal fuzzy stopping times are given under the assumptions of monotonicity and regularity for stopping rules also we find that fuzzy stopping times are favored in a comparison between fuzzy and classical stopping models

q scale measures of fuzzy sets abstract in an attempt to generalize nahmias scale measure to apply to fuzzy sets in this paper we introduce the notion of q scale measure a q scale measure is slightly different from a crisp valued possibility measure in several ways we use a number of examples to demonstrate the validity of the notion of q scale measure in various circumstances and give preliminary discussions on basic properties of q scale measure induced q scale measure semigroup of q scale measures q scale measure of probabilistic sets and q scale measure of l fuzzy sets

system failure engineering and fuzzy methodology an introductory overview abstract system failure engineering may encompass reliability safety security maintainability risk and quality control in this paper we pay a unified view i e a failure oriented view to system failure engineering the notion of failure can be represented in terms of fuzzy sets and widely interpreted we then discuss various application aspects of fuzzy methodology in system failure engineering which include fuzzy methodology in fault diagnosis in probist systems in structural reliability in software reliability in human reliability in safety engineering in security engineering in risk engineering and in quality control as well as profust posbist and posfust reliability theories engineering case studies and mathematical problems raised by applications of fuzzy methodology in system failure engineering are also addressed in this paper

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

a convergence theorem of fuzzy valued martingales in the extended hausdorff metric h in this paper we shall give a new embedding method to prove a convergence theorem for fuzzy valued random variables in the sense of the extended hausdorff metric h without the restriction of fuzzy sets satisfying the lipschitz condition

convergence theorems for fuzzy random variables and fuzzy martingales in this paper we discuss the properties of fuzzy random variables and fuzzy conditional expectation some extended results of dominated convergence theorems for fuzzy random variables are proved we define the concept of a right closed fuzzy martingale and give the necessary and sufficient conditions of right closability for fuzzy martingales the convergence theorem for fuzzy martingales is also extended

support functions for fuzzy sets the main purpose of this paper is to introduce and develop the notion of support functions for fuzzy sets this approach gives the opportunity to investigate a lot of properties of fuzzy valued functions functions with values in space of fuzzy sets defined on banach space using rich apparatus of functionals in this paper we examine some aspects of measurability integrability and measure theory

convergence of set valued sub and supermartingales in the kuratowski mosco sense the purpose of this paper is to prove some convergence theorems of closed and convex set valued sub and supermartingales in the kuratowski mosco sense to get submartingale convergence theorems we give sufficient conditions for the kudo aumann integral and hiai umegaki conditional expectation to be closed both for compact convex set valued random variables and for closed convex set valued random variables we also give an example of a bounded closed convex set valued random variable whose kudo aumann integral is not closed

fuzzy integrals based on pseudo additions and multiplications on definit un semi groupe ou un groupe qui est isomorphe au semi groupe ou du groupe ou la mesure de lebesque est definie on definit une integrale par pseudo addition et pseudo multiplication qui sont des operations de semi anneaux distributives et associatives

least squares fuzzy regression with fuzzy random variables let x y be two convex fuzzy random variables on rn using a suitable metric we prove that the conditional expectation e y x is the best approximation of y by measurable functions of x this generalizes the analogous and well known property for real random variables a further topic is the approximation of y by a linear function of x in special cases and by use of hukuhara s difference between fuzzy sets we obtain formulas which are analogous to the classical structure contrary to the classical fact however the conditional expectation of gaussian fuzzy random variables in general does not coincide with the linear regression function

an analytic characterization of the eigenvalues of self adjoint extensions let a be a self adjoint extension in k of a fixed symmetric operator a in k k an analytic characterization of the eigenvalues of a is given in terms of the q function and the parameter function in the krein naimark formula here k and k are krein spaces and it is assumed that a locally has the same spectral properties as a self adjoint operator in a pontryagin space the general results are applied to a class of boundary value problems with dependent boundary conditions

convergences of sequences of set valued and fuzzy set valued functions abstract we define some metrics on the space of integrably bounded multivalued functions and the space of integrably bounded fuzzy random variables with values in a separable banach space we also define various convergence of sequences of set valued and fuzzy set valued functions we investigate relationships among those convergences finally we obtain lebesguedominated convergence theorem for a sequence of fuzzy random variables

randomly weighted sums for exchangeable fuzzy random variables many statistical estimators are expressible in the form sn nk 1 ankxnk where ank are constants and xnk are fuzzy random variables then the convergence in probability or with probability one for sn is of interest in determining consistency of the estimators in statistical problems the linear form sn involves possibly random weights and possibly dependent fuzzy random variables which are generated from independent identically distributed fuzzy random variables thus we assume the sequence of fuzzy random variables xnk k 1 to be permutation invariant with respect to distributions instead of assuming independence in particular we look at the stochastic convergence of the form 1 an nk 1 anxnk which is slightly different from the initial form sn

approximating integrably bounded fuzzy random variables in terms of the generalized hausdorff metric abstract consider the class of the mappings from a euclidean space to the unit interval 0 1 that is the class of the fuzzy sets of this space which are upper semicontinuous the closure of their supports is compact and the inverse images of the singleton are nonempty sets when the metric d which generalizes the hausdorff metric is defined on the preceding class the resulting metric space is complete but nonseparable consequently for this nonseparability given a probability space and an integrably bounded fuzzy random variable associated with it one cannot ensure the existence of a sequence of simple fuzzy random variables d converging almost surely to the former one however we are going to show in this paper that for any given integrably bounded fuzzy random variable we can guarantee the existence of a sequence of integrably fuzzy random variables whose level functions are simple random compact sets and which d converges at every point to the former variable on the basis of this result we will obtain in this paper a constructive definition of integrably bounded fuzzy random variables and an operational way to compute their fuzzy expected values

a generalization of the minkowski embedding theorem and applications abstract puri and ralescu 1985 gave recently an extension of the minkowski embedding theorem for the class eln of fuzzy sets u on rn with the level application l u lipschitzian on the c 0 1 x sn 1 space in this work we extend the above result to the class ecn of level continuous applications moreover we prove that ecn is a complete metric space with e l n nb e c n and e l n e c n to prove the last result we use the multivalued bernstein polynomials and the vitali s approximation theorem for multifunction also we deduce some properties in the setting of fuzzy random variable multivalued

spectral analysis of singular ordinary differential operators with indefinite weights abstract in this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions we prove a general perturbation result on the local spectral properties of selfadjoint operators in krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares this result is applied to singular indefinite sturm liouville operators and higher order singular ordinary differential operators with indefinite weight functions

doob s stopping theorem for fuzzy super sub martingales with discrete time fuzzy random variables are introduced as random variables whose values are not reals but fuzzy sets the concepts of fuzzy conditional expectation and fuzzy super sub martingales are given in this paper we study doob s stopping theorem on fuzzy super sub martingales with discrete parameters

the strong law of large numbers for fuzzy random variables abstract sequences of independent and identically 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 variable

embedding of level continuous fuzzy sets on banach spaces abstract in this paper we present an extension of the minkowski embedding theorem showing the existence of an isometric embedding between the class f c x of compact convex and level continuous fuzzy sets on a real separable banach space x and c 0 1 b x the banach space of real continuous functions defined on the cartesian product between 0 1 and the unit ball b x in the dual space x also by using this embedding we give some applications to the characterization of relatively compact subsets of f c x in particular an ascoli arzela type theorem is proved and applied to solving the cauchy problem x t f t x t x t0 x0 on f c x

fuzzy variables as a basis for a theory of fuzzy reliability in the possibility context abstract conventional reliability theory is based on the binary state assumption and the probability assumption namely 1 the systems are precisely defined as functioning or failed 2 the system behavior is fully characterized in the context of probability measures however the probability assumption is no longer reasonable in the case of small size of samples since the term probability stems from the sample generality in this way possibility measures should be used to characterize the sample particularity in this paper the binary state assumption is reserved and the possibility assumption is taken in place of the probability assumption thus a concept of fuzzy reliability i e posbist reliability is introduced and a theory of fuzzy reliability i e posbist reliability theory is established various conclusions which are contrary to statements in conventional reliability theory i e probist reliability theory are then drawn

a note on indefinite integrals and absolute continuity for fuzzy valued mappings abstract this note is to define the absolute continuity for fuzzy valued mappings and to give the necessary and sufficient conditions for the newton leibniz formula to hold

gaussian processes and martingales for fuzzy valued random variables with continuous parameter abstract the purpose of this paper is to discuss fuzzy valued gaussian processes and martingales with continuous parameter we mainly obtain a representation theorem for fuzzy valued gaussian processes after proving a new embedding theorem based on previous results about set valued and fuzzy valued martingales cf s li y ogura j fuzzy math 4 1996 905 s li y ogura fuzzy sets and systems 101 1999 453 s li y ogura ann probab 26 1998 1384 1402 s li y ogura in proc ifsa 97 vol 4 1997 pp 9 13 s li y ogura in c bertoluzza m a gil d a ralescu eds statistical modeling analysis and management of fuzzy data physica in press y ogura s li fuzzy sets and systems in press we prove optional sampling theorems for set valued and fuzzy valued martingales with continuous parameter

convexification in limit laws of random sets in banach spaces a strong law of large numbers for random compact sets in a banach space with the minkowski addition is provided by puri and ralescu 6 and among other results by gine hahn and zinn 2 and hess 3 this follows the finite dimensional version introduced in 1 the latter however allows nonconvex sets while the generalizations to the banach space setting were proved for convex sets a convexification argument stated in the lemma below shows that the convexity assumption can be dropped the proof reveals information about the nature of the convergence since writing this note we learned that f hiai 4 has independently established this result using a different method some notations let b be a banach space with norm 1111 the minkowski addition of two compact subsets k and l is k l x y x e k y e li and ak is the set tax x e k the closed convex hull of k is denoted cok it is a linear operation the hausdorff distance h k l is defined as max maxxekminyel 11 x y 11 maxyelminxek 11 x y y convergence of sets in this paper is meant in the hausdorff metric

limit theorems for fuzzy random variables a strong law of large numbers and a central limit theorem are proved for independent and identically distributed fuzzy random variables whose values are fuzzy sets with compact levels the proofs are based on embedding theorems as well as on probability techniques in banach space