Shoumei Li

Limit theorems and applications of set-valued and fuzzy set-valued random variables

Preface. Part I: Limit Theorems of Set-Valued and Fuzzy Set-Valued Random Variables. 1. The Space of Set-Valued Random Variables. 2. The Aumann Integral and the Conditional Expectation of a Set-Valued Random Variable. 3. Strong Laws of Large Numbers and Central Limit Theorems for Set-Valued Random Variables. 4. Convergence Theorems for Set-Valued Martingales. 5. Fuzzy Set-Valued Random Variables. 6. Convergence Theorems for Fuzzy Set-Valued Random Variables. 7. Convergences in the Graphical Sense for Fuzzy Set-Valued Random Variables. References for Part I. Part II: Practical Applications of Set-Valued Random Variables. 8. Mathematical Foundations for the Applications of Set-Valued Random Variables. 9. Applications to Imaging. 10. Applications to Data Processing. References for Part II. Index.

Representation theorems, set-valued and fuzzy set-valued Ito integral

In this paper, we shall present representation theorems of set-valued martingales and set-valued processes of finite variation with continuous time. We shall also obtain a representation theorem of a predictable set-valued stochastic process. We shall give a new definition of Ito integral of a set-valued stochastic process with respect to a Brownian motion based on the work [E.J. Jung, J.H. Kim, On set-valued stochastic integrals, Stochastic Anal. Appl. 21(2) (2003) 401-418.]. We shall also discuss some properties of set-valued Ito integral, especially the presentation theorem of set-valued Ito integral. Finally, we extend some of above results to the fuzzy set-valued case.

Central limit theorems for generalized set-valued random variables

Abstract We give central limit theorems for generalized set-valued random variables whose level sets are compact both in R d or in a Banach space under milder conditions than those obtained recently by the latter two authors.

Strong laws of large numbers for independent fuzzy set-valued random variables

Abstract In this paper, we shall present strong laws of large numbers (SLLNs) for independent (not necessary identically distributed) fuzzy set-valued random variables whose base space is a separable Banach space or an Euclidean space, in the sense of the extended Hausdorff metric d H ∞ . We apply the method to the sequence of independent identically distributed fuzzy set-valued random variables to give a simple proof of SLLNs.

Set defuzzification and choquet integral

In this paper, we discuss the defuzzification problem. We first propose a set defuzzification method, (from a fuzzy set to a crisp set) by using the Aumann integral. From the obtained set to a point, we have two methods of defuzzification. One of these uses the mean value method and the other uses a fuzzy measure. In the first case, we compare our mean value method with the method of the center of gravity. In the second case, we compare fuzzy measure method with the Choquet integral method. We also give there a sufficient condition so that the results in the last two methods are equivalent.

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.

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 Limit Theorems for Random Fuzzy Sets Including Large Deviation Principles

Study of random fuzzy sets or fuzzy set-valued random variables was initiated by Feron [7] and Kwakernaak [13, 14] in late 70’s, about ten years after the famous paper by Zadeh [32]. A systematic treatment of them was done by Puri and Ralescu [30] in the case when the underlying space is ℝd. For the general underlying space, see Li and Ogura [15].

Large and Moderate Deviations of Random Upper Semicontinuous Functions

We firstly discuss the topological properties of the space of upper semicontinuous functions, and then we obtain large deviation principles for random upper semicontinuous functions under various topologies. Finally, we prove moderate deviation principles for random sets and random upper semicontinuous functions.

Convergence in graph for fuzzy valued martingales and smartingales

In this paper, we introduce the concept of convergence in graph for fuzzy-valued random variables, give an equivalent definition and then obtain convergence theorems for fuzzy-valued martingales, submartingales and supermartingales based on the results of our previous papers (Li and Ogura, 1996, 1998, 1999).