Study on Limit Theorems for Random Sets with Dependency

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For random sets(or fuzzy random sets) being dependent limit theorems were obtained under the condition that random sets are not compact.Many of the results of laws of large numbers for Banach spaced-value random setswere assumed to be IID(Independent and Identical Distribution) with compactness, and proved by the embedding method and its extension.Moreover, it will be realistic in establishing limit theorems to put some dependency among random sets though IID is, in general, assumed.In this research(1)the author proved strong laws of large numbers for random sets(fuzzy random sets)which posses the nature of dependency, in particular, exchangeability.The topology is based on Kuratowski-Mosco convergence which is weaker than Hausdorff sense and fits into the case of unbounded random sets with more applicable nature.(2)One algorithm to solve convex maximization problem was established with fuzzy constraints.The objective function of encountered problem is convex and its feasible region is inverse convex. In the future work, it is expected to establish algorithms of maximization problems by introducing random sets concept with stochastic process of fuzzy nature.