Random Locations of Periodic Stationary Processes

TL;DR

This paper characterizes the possible distributions of intrinsic location functionals in periodic stationary processes, revealing their convex structure and exploring subclasses with uniform density bounds and joint mixability properties.

Contribution

It provides a complete description of the distribution set for intrinsic location functionals and analyzes specific subclasses with novel properties.

Findings

The set of all distributions is the convex hull of a specific group.

Subclasses include those with densities bounded below by a positive constant.

Distributions are related to the concept of joint mixability.

Abstract

We consider a family of random locations, called intrinsic location functionals, of periodic stationary processes. This family includes but is not limited to the location of the path supremum and first/last hitting times. We first show that the set of all possible distributions of intrinsic location functionals for periodic stationary processes is the convex hull generated by a specific group of distributions. We then focus on two special subclasses of these random locations. For the first subclass, the density has a uniform lower bound; for the second subclass, the possible distributions are closely related to the concept of joint mixability.