Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables

In this text we shall consider the spaces formed by all closed or compact (convex) subsets of a Banach space. We shall call such spaces hyperspaces. Throughout this book, assume that (X, ║ · ║ _X )is a Banach space with the dual space X*, 0 denotes the origin of X, θ denotes the. one element set {0} and ℝ is the set of all of real numbers. The open unit ball in X is denoted by U. Also we shall adopt particular notation for certain classes of subsets of X, which we call hyperspaces.

Throughout this chapter we shall assume that (Ω, A, µ) is a finite measure space for simplicity, although most of the results are valid for σ-finite measure space. For a set-valued random variable F we denote by S _F its selection set S1/F, in L ¹[Ω; K(X)].

The strong law of large numbers was first proved by Artstein and Vitale (cf. [11]) for independent identically distribution random variables whose values are compact subsets of finite-dimensional Euclidean space ℝ ^d . This strong law with Hausdorff metric convergence has been rewritten or extended by several authors such as Cressie [50], Giné, Hahn and Zinn [72], Hess [82], Hiai [90], Puri and Ralescu [177], Taylor and Inoue [203]. In this section we shall first prove a strong law of large numbers for independent identically distribution random variables whose values are compact convex subsets of a separable Banach space X, and then drop convexity for the compact-valued case. We shall also discuss the case with independence only. We first give some necessary notations.

Set-valued martingales were first introduced by Van Cutsem in 1969 (cf. [212]) in the case of convex compact valued ones. This is a natural extension of the theory of martingales in a Banach space. After this work the theory of the set-valued martingale was studied by several authors such as Neveu [157], Daures [54]. However, owing to the limit of the definition of conditional expectation defined by Van Cutsem the study was restricted to the case of convex compact-valued sets. In 1977 Hiai and Umegaki gave a new definition of conditional expectations by using the selection method, as we have seen in Section 3 of Chapter 2, for the closed set-valued random variables in a Banach space. This made the study of set-valued martingales more extensive and deep. For example, a representation theorem of set-valued martingales was proved by Luu by means of martingale selections [141]. Convergence theorems of martingales, sub- and supermartingales under various settings were obtained by many authors, Korvin and Kleyle [119], Papageorgiou [169, 170, 171], Hess [85], Wang and Xue [218], Li and Ogura [131] and [133]. Also there are many more works of extension toward set-valued quasi-martingales, asymptotic martingales, etc.. In this Chapter we shall focus on the discussion of the basic concepts and of convergence theorems for martingales, sub- and supermartingales.

In practice we are often faced with random experiments whose outcomes are not numbers but are expressed in inexact linguistic terms. For example, consider a group of individuals chosen at random who are questioned about the weather in a particular city on a particular winter’s day. The resulting data of this random experiment would be linguistic terms such as ‘cold’, ‘more or less cold’, ‘very cold’, ‘extremely cold, which can be described by fuzzy sets, introduced by Zadeh in 1965 in his paper [234], rather than by a single real number or subsets of real numbers. A natural question which arises with reference to this example is: what is the average opinion about the weather in that particular city on a particular day? A possible way of handling ‘data’ like this is by using the concepts of fuzzy sets and expectations of fuzzy set-valued random variables. Fuzzy set-valued random variables are random variables whose values are not numbers or sets but fuzzy sets.

In this Chapter we also focus on the case of the basic space X = ℝ ^d , although many results can be obtained in the case in which the basic space is a general Banach space, in particular, for a reflexive Banach space or for X* being separable.

In this section we shall introduce two convergences in the graphical sense, i.e., convergences in the graphical Kuratowski—Mosco sense and in the graphical Hausdorff sense. When we discuss the convergence in the Hausdorff metric we may assume that the basic space X is only a metric space. When we discuss the Kuratowski—Mosco convergence we have to assume that the basic space X is a Banach space since it is related to weakly convergence. Thus we shall state that X is a metric space or a Banach space, respectively, in the following theorems.

In the previous seven chapters we have presented many complex mathematical results. To mathematicians these results themselves are of interest, and this mathematical interest justifies our research. If these results were about some very abstract, far from practical mathematical concepts then this purely mathematical interest would probably be our only reward. However, our results are not about some very abstract mathematical concepts, these results are about set-valued random variables, a natural concept from probability theory. Probability theory started as an analysis of real world random processes and events — and continues to be a foundation for this analysis. Results about (number-valued) random variables and function-valued random variables (random processes) are actively applied in all areas of engineering and science. It is therefore natural to expect that results about random sets should also have numerous practical applications. This expectation is indeed correct. In this and following chapters we will show that limit theorems about set valued random variables can be used to solve numerous practical problems.

With this chapter we start describing applications of our set based techniques. Since set theory is the main language of the foundations of mathematics it is natural to expect that set methods can be applied in many application areas — and, as we shall see, symmetry methods related to limit theorems are indeed useful in multiple application areas.

What is the set of possible values of a measurement error? In the majority of practical applications an error is caused not by a single cause; it is caused by a large number of independent causes, each of which adds a small component to the total error. This fact is widely used in statistics: namely, since it is known that the distribution of the sum of many independent small random variables is close to one of the so called infinitely divisible ones (a class which includes the well known Gaussian distribution), we can safely assume that the distribution of the total error is infinitely divisible. This assumption is used in the majority of the statistical applications.