Study on Limit Theorems for Random Sets with Dependency

对于随机集（或模糊随机集）是相依的，在随机集不紧的条件下得到了极限定理。将Banach空间值随机集的许多大数定律的结果假定为具有紧性的IID（独立相同分布），并通过嵌入方法及其推广加以证明。此外，尽管IID通常是假设的，但在建立极限定理时，在随机集之间放置一些依赖关系将是现实的。在本研究(1)中，作者证明了随机集（模糊随机集）具有相依性，特别是互换性的强大数定律。该拓扑基于Kuratowski-Mosco收敛性，该收敛性比Hausdorff感觉弱，适合无界随机集的情况，具有更强的适用性。(2)建立了一种求解模糊约束条件下凸最大化问题的算法。所遇到问题的目标函数是凸的，可行域是逆凸的。在未来的工作中，期望通过引入具有模糊性质的随机过程的随机集概念来建立最大化问题的算法。

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.