面向集值与模糊集值随机变量的极限理论及其应用

This project will consider the limit theory of set-valued random variables and random upper semicontinuous functions. In the real life, due to a lot of factors such as the missing information and limitation of people's realization, set-valued random variables and random upper semicontinuous functions can better describe this kind of uncertainty, so our subject is set-valued random variables and random upper semicontinuous functions which is a kind of extension of real-valued random variables and Banach-valued random variables. In the limit theory of set-valued random variables, the law of large numbers and the central limit theory have been completely researched, we will consider those limit theories which are more precise than the law of large numbers and the central limit theory, that is, large deviations and moderate deviations, thereby we can better develop the limit theory of set-valued random variables. According to present research status, our project will mainly complete 4 parts: (1)moderate deviations of bounded closed convex random sets and bounded closed convex random upper semicontinuous functions; (2)functional large deviations of compact random sets and compact random upper semicontinuous functions;(3)using large deviations of compact random sets, we will solve the large deviations of real-valued random variables under non-additive measures, which can provide a new method for the limit theory of non-additive measure; (4)we will apply functional large deviations of compact random sets into mobile opportunistic networks, and solve the asymptotic behavior of the distribution of the first passage into a given field.

本课题所研究的是集值和模糊集值随机变量序列极限理论方面的内容。由于现实生活中信息的缺失以及人们认识问题的局限性等因素，集值或模糊集值能更好的、更合理地描述过程中的这种不确定性，故本课题研究的对象是以集合和模糊集为值的随机变量，它是取值为实数或Banach值的单点值情况的推广。在集值理论中，大数定律及中心极限定理已被研究得较为完善，本课题欲考虑比以上极限定理更为精细的行为，即大偏差原理与中偏差原理，从而更好的发展集值极限理论的内容。根据国内外研究现状，本项目主要完成四个内容：(1)有界闭凸集值和模糊集值随机变量的中偏差；(2)紧集值和模糊集值随机变量的泛函形式的大偏差；(3)利用集值随机变量序列的大偏差问题解决非可加测度下实值随机变量序列的大偏差问题，为非可加测度的极限理论研究提供新的方法；(4) 将集值随机变量的轨道大偏差结论应用到移动机会网络，解决消息首次传入目标区域的分布的渐近行为。

由于现实生活中信息的缺失以及人们认识问题的局限性等因素，集值或模糊集值能更好的、更合理地描述过程中的这种不确定性，故本课题研究的对象是以集合和模糊集为值的随机变量。一方面通过研究集值理论的大偏差问题对经典的大偏差理论提出了新的挑战。另一方面可以完善集值理论并将理论应用于其他场景。本项目主要考虑了四个内容:(1)有界闭凸集值和模糊集值随机变量的中偏差;(2)紧集值和模糊集值随机变量的泛函形式的大偏差;(3)利用集值随机变量序列的大偏差问题解决非可加测度下实值随机变量序列的大偏差问题，为非可加测度的极限理论研究提供新的方法;(4)将集值随机变量的轨道大偏差结论应用到移动机会网络，解决消息首次传入目标区域的分布的渐近行为。目前我们已获得如下结果:(i)独立同分布区间值随机变量的样本均值落在一个闭区间里的概率的上界估计;(ii) 紧集值和模糊集值随机变量满足泛函形式的大偏差;(iii) 非可加测度下实值随机变量序列的大偏差上界已获得，下界还有待进一步的研究;(iv)移动机会网络获得了较多的成果，对于消息按不同的随机过程进行传播时得到了消息首次传入目标区域的分布的渐近行为，为消息传播的推进或控制给出了很好的理论依据。