Study on limit theorems for fuzzy set-valued random variables

1.给出了一类具有连续路径的一维马尔可夫(不一定是强马尔可夫)过程为双广义扩散过程的充要条件。值得注意的是，我们这类过程的尺度函数一般不再是连续的，在尺度的不连续点处可能会遗漏强马氏性。给出了具有紧水平集的独立同分布模糊集值随机变量和的大偏差原理。更确切地说，对于由p阶的平均收敛、Levy度量和图收敛引起的拓扑以及可数直积拓扑的相对拓扑，我们给出了这类和的Cramer型大偏差原理。我们把关于集值随机变量构造的Choquite定理推广到关于模糊集值随机变量构造的Choquite定理。也就是说，我们证明了一个模糊集值随机变量的存在性，该模糊集值随机变量对应于与乘积Fall-Matheron拓扑的相对拓扑重合的图拓扑中的无穷阶交错Choquite容量。在局部紧的第二可数Hausdorff空间中，给出了闭集空间上与Fall-Matheron拓扑相容的具体度量。我们不得不修改I.Molchanov在《随机集》一书中给出的Hausdorff-Buseman，它是作为相容度量介绍的。我们给出了独立模糊集值随机变量和的一个强大数定律，它们不一定是渐近分布的。

1. We gave a necessary and sufficient condition for stochastic processes in a class of one-dimensional Markovian (not necessarily strong Markovian) processes with continuous paths to be bi-generalized diffusion process which were introduced by the head investigator. It is worth to note that the scale functions of the processes in our class are no more continuous in general and may miss the strong Markov property at the discontinuous points of scales.2. We gave large deviation principles for sums of independent identically distributed fuzzy set-valued random variables with compact level sets. More precisely, we gave Cramer type large deviation principles for such sums with respect to the topologies induced by the mean convergence of order p, Levy's metric and graph convergence, and the relative topology of a countable direct product topology.3. We extended a Choquet theorem on the construction of set-valued random variables to that on the construction of fuzzy set-valued random variables. That is, we showed existence of a fuzzy set-valued random variable associated with an alternating Choquet capacity of infinite order with respect to the graph topology which coincides the relative topology of product Fell-Matheron topology.4. We gave a concrete metric compatible with the Fell-Matheron topology on the space of closed sets in a locally compact second countable Hausdorff space. We had to modify Hausdorff-Buseman which is given in the book "Random sets" by I. Molchanov introduced as a compatible metric.5. We gave a strong law of large numbers for sums of independent fuzzy set-valued random variables, which are not necessarily indentically distributed.