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.