Random locations, ordered random sets and stationarity

Intrinsic location functional is a large class of random locations closely related to stationary processes. In this paper the author firstly identifies a subclass of intrinsic location functional and proves that it characterizes stationary increment processes. Then a generalization of intrinsic location functional is introduced and its relationship with intrinsic location functional is discussed. Finally we develop representation results using ordered random sets and piecewise linear functions. It is proved that each random location corresponds to the maximal element in a random set according to certain order, and the locations change in a specific way when the path is translated.