On multiple random locations of stationary processes

Abstract

We generalize the concept of intrinsic location functionals to accommodate n=2 random locations, which we combine together in either sets or vectors. For the set-valued case of "intrinsic multiple-location functionals" we show that for any stationary process and compact interval, the distribution of any intrinsic multiple-location functionals is absolutely continuous on the interior of the interval, the density exists everywhere, is càdlàg, bounded at each point of the interior and satisfies certain total variation constraints. We also characterize the class of possible distributions, showing that it is a weakly closed compact set, and we find its extreme points. Moreover, we show that for almost every measure m in this class of distributions one can construct a pair comprising a stationary process and intrinsic multiple-location functional which has m as its distribution. For the vector-valued case of "intrinsic location vectors", we identify subclasses based on the joint behaviour of the two random locations and derive results for each subclass. Some of the results connect the intrinsic location vectors back to the "single-location case" of intrinsic location functionals.