Study on limit theorems for fuzzy set-valued random variables

1. 本文给出了一类具有连续路径的一维马尔可夫过程（不一定是强马尔可夫过程）的随机过程是双广义扩散过程的充要条件。值得注意的是，我们类中过程的尺度函数通常不再是连续的，并且可能在尺度的不连续点上错过强马尔可夫性质。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。更准确地说，我们给出了关于p阶平均收敛、Levy的度量和图收敛以及可数直积拓扑的相对拓扑所引起的拓扑的Cramer型大偏差原理。将集值随机变量构造的Choquet定理推广到模糊集值随机变量构造的Choquet定理。也就是说，我们证明了在与乘积Fell-Matheron拓扑的相对拓扑相一致的图拓扑上，存在一个与无限阶交替Choquet容量相关的模糊集值随机变量。给出了局部紧第二可数Hausdorff空间中闭集空间上与Fell-Matheron拓扑相容的一个具体度量。我们必须修改由I. Molchanov在《随机集》一书中作为兼容度量引入的Hausdorff-Buseman。我们给出了独立模糊集值随机变量和的一个强大数定律，这些变量不一定是同分布的。

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.