随机位置、有序随机集与平稳性
Random Locations, Ordered Random Sets and Stationarity
摘要：
内在位置泛函是一大类随机位置，包含在许多情况下可能遇到的位置，例如，路径上确界/下确界在给定区间上的位置，第一个/最后一个首中时等。它已被证明，这个概念是非常密切相关的平稳随机过程，并可用于表征平稳性。本文首先确定了内禀位置泛函的一个子类，并证明了这个子类与平稳增量过程有着密切的关系。然后我们用随机偏序点集和分段线性函数描述了内禀位置泛函。证明了该类中的每一个随机位置都对应于区间上随机集中的极大元的位置，且按一定的偏序。此外，当感兴趣的间隔沿着实线移动时，位置以非常特定的方式改变。在此基础上，引入了一种广义的内禀位置泛函，称为“局部内禀位置泛函”，并研究了它与内禀位置泛函的关系。
英文摘要：
Intrinsic location functional is a large class of random locations containing locations that one may encounter in many cases, e.g., the location of the path supremum/infimum over a given interval, the first/last hitting time, etc. It has been shown that this notion is very closely related to stationary stochastic processes, and can be used to characterize stationarity. In this paper the author firstly identifies a subclass of intrinsic location functional and proves that this subclass has a deep relationship to stationary increment processes. Then we describe intrinsic location functionals using random partially ordered point sets and piecewise linear functions. It is proved that each random location in this class corresponds to the location of the maximal element in a random set over an interval, according to certain partial order. Moreover, the locations changes in a very specific way when the interval of interest shifts along the real line. Based on these ideas, a generalization of intrinsic location functional called "local intrinsic location functional" is introduced and its relationship with intrinsic location functional is investigated.
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