A Note on Occupation Times of Stationary Processes

Abstract

Consider a real valued stationary process $X=\{X_s : s\in \mathbb{R}\}$. For a fixed $t\in \mathbb{R}$ and a set $D$ in the state space of $X$, let $g_t$ and $d_t$ denote the starting and the ending time, respectively, of an excursion from and to $D$ (straddling $t$). Introduce also the occupation times $I^+_t$ and $I^-_t$ above and below, respectively, the observed level at time $t$ during such an excursion. In this note we show that the pairs $(I^+_t, I^-_t)$ and $(t-g_t, d_t-t)$ are identically distributed. This somewhat curious property is, in fact, seen to be a fairly simple consequence of the known general uniform sojourn law which implies that conditionally on $I^+_t + I^-_t = v$ the variable $I^+_t$ (and also $I^-_t$) is uniformly distributed on $$. We also particularize to the stationary diffusion case and show, e.g., that the distribution of $I^-_t+I^+_t$ is a mixture of gamma distributions.