Intrinsic location functionals of stationary processes

Abstract

We consider a large family of measurable functionals of the sample path of a stochastic process over compact intervals (including first hitting times, leftmost location of the supremum, etc.) we call intrinsic location functionals. Despite the large variety of these functionals and their different nature, we show that for stationary processes the distribution of any intrinsic location functional over an interval is absolute continuous in the interior of the interval, and the density functions always have a version satisfying the same total variation constraints. Conversely, these total variation constraints are shown to actually characterize stationarity of the underlying stochastic process. We also show that the possible distributions of the intrinsic location functionals over an interval form a weakly closed convex set and describe its extreme points, and present applications of this description.