Is the location of the supremum of a stationary process nearly uniformly distributed?

TL;DR

This paper investigates the distribution of the location of the supremum in stationary processes, revealing it is not always uniform and providing conditions and bounds for its density.

Contribution

It characterizes the density of the supremum's location, proving absolute continuity, establishing bounds, and identifying conditions for the distribution in stationary processes.

Findings

Supremum location distribution is absolutely continuous inside the interval.

Universal upper bounds on the density are established and shown to be optimal.

The distribution can deviate from uniformity under certain conditions.

Abstract

It is, perhaps, surprising that the location of the unique supremum of a stationary process on an interval can fail to be uniformly distributed over that interval. We show that this distribution is absolutely continuous in the interior of the interval and describe very specific conditions the density has to satisfy. We establish universal upper bounds on the density and demonstrate their optimality.