Random Locations, Ordered Random Sets and Stationarity

TL;DR

This paper explores the relationship between intrinsic location functionals, stationary processes, and stationary increment processes, introducing a subclass and a generalized concept called local intrinsic location functional.

Contribution

It identifies a subclass of intrinsic location functionals linked to stationary increment processes and introduces local intrinsic location functionals, expanding the theoretical framework.

Findings

Characterization of intrinsic location functionals via random partially ordered sets

Deep relationship established between a subclass of intrinsic location functionals and stationary increment processes

Introduction of local intrinsic location functional and analysis of its properties

Abstract

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.