Study on limit theorems for fuzzy set-valued random variables

点击翻译按钮获取中文摘要

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.