Electronic Communications in Probability

A Note on Occupation Times of Stationary Processes

Marina Kozlova and Paavo Salminen

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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.

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Electron. Commun. Probab., Volume 10 (2005), paper no. 10, 94-104.

Accepted: 9 June 2005
First available in Project Euclid: 4 June 2016

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Kozlova, Marina; Salminen, Paavo. A Note on Occupation Times of Stationary Processes. Electron. Commun. Probab. 10 (2005), paper no. 10, 94--104. doi:10.1214/ECP.v10-1138. https://projecteuclid.org/euclid.ecp/1465058075

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