Sojourns and Extremes of Stationary Processes

TLDR

The paper studies a real stationary stochastic process with continuous sample paths, focusing on the Lebesgue measure of exceedances \(L_t(u)\) and the running maximum \(M(t)\), and reviews prior results on their limiting behavior for Gaussian processes under two asymptotic regimes. The paper aims to extend the Gaussian‑case methods to general, non‑Gaussian stationary processes. The extension is illustrated through applications to specific non‑Gaussian examples and classes that include Gaussian subclasses.

Abstract

Let $X(t), -\infty < t < \infty$, be a real stationary stochastic process with continuous sample functions. For $t > 0$, put $L_t(u) =$ Lebesgue measure of $\{s: 0 \leq s \leq t, X(s) > u\}$ and $M(t) = \max(X(s): 0 \leq s \leq t)$. For several years the author has studied the limiting properties of these random variables in the case where $X(t)$ is a Gaussian process and under two kinds of limiting operations: i) $t$ fixed and $u \rightarrow \infty$; ii) $t \rightarrow \infty$ and $u = u(t) \rightarrow \infty$ as a function of $t$. The purpose of this paper is to show how the methods developed in the Gaussian case can be extended to the general, not necessarily Gaussian case. This is illustrated by applications of some of the results to specific examples of non-Gaussian processes, and classes of processes containing a Gaussian subclass.