Mathematical and Statistical Methods: Volume 43 This book presents a clear, systematic treatment of convergence theorems of set-valued random variables (random sets) and fuzzy set-valued random variables (random fuzzy sets). Topics such as strong laws of large numbers and central limit theorems, including new results in connection with the theory of empirical processes are covered. The author's own recent developments on martingale convergence theorems and their applications to data processing are also included. The mathematical foundations along with a clear explanation such as H

The purpose of this paper is to prove some convergence theorems of closed and convex set valued sub-and super-martingales in the Kuratowski-Mosco sense. For getting submartingale convergence theorem, we give sufficient conditions for 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.

The purpose of this paper is to prove the convergence thorems of set valued and fuzzy valued martingales in Kuratowski-Mosco sence based on [4].

We prove that the graph convergence in Hausdorff metric or Kuratowski-Mosco topology of a sequence of fuzzy sets follows from the conrgence 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.

The purpose of this paper is to discuss fuzzy valued Gaussian processes and mar-tingales 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 theo- rems for set valued and fuzzy valued martingales with continuous parameter.

In this paper, we discuss the defuzzification problem. We first propose a set defuification 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.

In this paper, we introduce the concept of convergence in graph for fuzzy val-ued 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 [14, 16, 17].

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

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 Haus-dorff metric H∞, without the restriction of fuzzy sets satisfying the Lipschitz condition.