@article{ECP1138,
author = {Marina Kozlova and Paavo Salminen},
title = {A Note on Occupation Times of Stationary Processes},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {10},
year = {2005},
keywords = {},
abstract = {Consider a real valued stationary process $X={X_s:, s\in R}$. For a fixed $t\in R$ 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 $(0,v)$. 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.},
pages = {no. 10, 94-104},
issn = {1083-589X},
doi = {10.1214/ECP.v10-1138},
url = {http://ecp.ejpecp.org/article/view/1138}}