Distribution of the supremum location of stationary processes

TL;DR

This paper characterizes all possible distributions of the supremum location in stationary processes and shows that, under strong mixing conditions, this distribution approaches uniformity as the interval length grows.

Contribution

It provides a comprehensive description of supremum location distributions for broad stationary processes and establishes conditions under which these distributions become uniform.

Findings

Supremum location distributions are fully characterized for broad classes of stationary processes.

Under strong mixing, the distribution of the supremum location converges to uniform as interval length increases.

The results unify understanding of supremum location behavior across different stationary processes.

Abstract

The location of the unique supremum of a stationary process on an interval does not need to be uniformly distributed over that interval. We describe all possible distributions of the supremum location for a broad class of such stationary processes. We show that, in the strongly mixing case, this distribution does tend to the uniform in a certain sense as the length of the interval increases to infinity.